5. Q5
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Similar Question 1
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p> <ul> <li></li> </ul>
Similar Question 2
<p>The fuse of a three-break firework rocket is programmed to ignite three times with 2-second intervals between the ignitions. When the rocket is shot vertically in the air, its height <code class='latex inline'>\displaystyle h </code> in feet after <code class='latex inline'>\displaystyle t </code> seconds is given by the formula <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+70 t </code>. At how many seconds after the shot should the firework technician set the timer of the first ignition to make the second ignition occur when the rocket is at its highest point? <code class='latex inline'>\displaystyle \begin{array}{lll}\text { A } 3 & \text { B } 9 & \text { C } 5\end{array} </code></p>
Similar Question 3
<p>The manager of a hardware store knows that the weekly revenue function for batteries sold can be modelled with <code class='latex inline'>R(x)= -x^2 + 10x + 30 000</code>, where both the revenue, <code class='latex inline'>R(x)</code>, and the cost, <code class='latex inline'>x</code>, of a package of batteries are in dollars. According to the model, what is the maximum revenue the store will earn?</p>
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Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
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<p>A company that manufactures MP3 players uses the relation <code class='latex inline'>P=120x-60x^2</code> to model its profit. The variable <code class='latex inline'>x</code> represents the number of thousands of MP3 players sold. The variable <code class='latex inline'>P</code> represents the profit in thousands of dollars.</p> <ul> <li>How many MP3 players must be sold to earn this profit?</li> </ul>
<p>A cliff diver dives from about <code class='latex inline'>17</code> m above the water. The divers height above the water. <code class='latex inline'>h(t)</code>,in metres, after <code class='latex inline'>t</code> seconds is modelled by <code class='latex inline'>h(t)=-4.9t^2+1.5t+17</code>. Explain how to determine when the diver is <code class='latex inline'>5</code> m above the water.</p>
<p>A football is thrown into the air. The height, <code class='latex inline'>\displaystyle h(t) </code>, of the ball, in metres, after <code class='latex inline'>\displaystyle t </code> seconds is modelled by <code class='latex inline'>\displaystyle h(t)=-4.9(t-1.25)^{2}+9 </code></p><p>a) How high off the ground was the ball when it was thrown?</p><p>b) What was the maximum height of the football?</p><p>c) How high was the ball at <code class='latex inline'>\displaystyle 2.5 \mathrm{~s} </code> ?</p><p>d) Is the football in the air after <code class='latex inline'>\displaystyle 6 \mathrm{~s} </code> ?</p><p>e) When does the ball hit the ground?</p>
<p>Use factoring to solve the following problem: The height of a ball above the ground is given by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+45 t+50 </code>, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. When will the ball hit the ground?</p>
<ol> <li>Football During a field goal attempt, the function <code class='latex inline'>\displaystyle h=-0.02 d^{2}+0.9 d </code> models the height, <code class='latex inline'>\displaystyle h </code> metres, of a football in terms of the horizontal distance, <code class='latex inline'>\displaystyle d </code> metres, from where the ball was kicked. Find the horizontal distance the ball travels until it first hits the ground.</li> </ol>
<p>A video tracking device recorded the height, <code class='latex inline'>h</code>, in metres, of a baseball after it was hit. The data collected can be modelled by the relation <code class='latex inline'>h=-5(t-2)^2+21</code>, where <code class='latex inline'>t</code> is the time in seconds after the ball was hit.</p> <ul> <li>When did the baseball reach its maximum height?</li> </ul>
<p>The average ticket price at a regular movie theatre (all ages) from 2015 to 2019 can be modelled by <code class='latex inline'>C = 0.06t^2 - 0.27t + 5.36</code>, where <code class='latex inline'>C</code> is the price in dollars and t is the number of years since 2015 ( <code class='latex inline'>t = 0</code> for 2015, <code class='latex inline'>t = 1</code> for 2016, and so on).</p><p><strong>a)</strong> When were ticket prices the lowest during this period?</p><p><strong>b)</strong> What was the average ticket price in 2018?</p><p><strong>c)</strong> What does the model predict the average ticket price will be in 2030?</p><p><strong>d)</strong> Write the equation for the model in vertex form.</p>
<p>The Wheely Fast Co. makes custom skateboards for professional riders. The company models its profit with the function <code class='latex inline'>\displaystyle P(b)=-2 b^{2}+14 b-20 </code>, where <code class='latex inline'>\displaystyle b </code> is the number of skateboards produced, in thousands, and <code class='latex inline'>\displaystyle P(b) </code> is the company&#39;s profit, in hundreds of thousands of dollars.</p><p>a) How many skateboards must be produced for the company to break even?</p><p>b) How many skateboards does Wheely Fast Co. need to produce to maximize profit?</p>
<p>An electronics store sells an average of 52 laptops per month at an average selling price that is <code class='latex inline'>\displaystyle \$ 660 </code> more than the cost price. For every <code class='latex inline'>\displaystyle \$ 40 </code> increase in the selling price, the store sells two fewer laptops. What</p><p>amount over the cost price will maximize revenue?</p>
<p>On Mars, if you hit a baseball, the height of the ball at time <code class='latex inline'>t</code> would be modelled by the quadratic function <code class='latex inline'>h(t)=-1.85t^2+20t+1</code>, where <code class='latex inline'>t</code> is in seconds and <code class='latex inline'>h(t)</code> is in metres.</p><p>How long will the ball be above <code class='latex inline'>17</code> m?</p>
<p>Fundraising Your class is selling boxes of flower seeds as a fundraiser. The total profit <code class='latex inline'>\displaystyle p </code> depends on the amount <code class='latex inline'>\displaystyle x </code> that your class charges for each box of seeds. The equation <code class='latex inline'>\displaystyle p=-0.5 x^{2}+25 x-150 </code> models the profit of the fundraiser. What&#39;s the smallest amount, in dollars, that you can charge and make a profit of at least <code class='latex inline'>\displaystyle \$ 125 ? </code></p>
<p>The sum of two numbers is 16. What is the largest possible product between these numbers?</p>
<p>A parabola has equation <code class='latex inline'>y = x^2 -3x - 18</code>.</p><p>a) Factor the right side of the equation.</p><p>b) Identify the x-intercepts of the parabola.</p><p>c) Find the equation of the axis of symmetry</p>
<p>A ball is kicked into the air. It follows a path given by <code class='latex inline'>\displaystyle h(t)=-4.9 t^{2}+8 t+0.4 </code>, where <code class='latex inline'>\displaystyle t </code> is the time, in seconds, and <code class='latex inline'>\displaystyle h(t) </code> is the height, in metres.</p><p>a) Determine the maximum height of</p><p>the ball to the nearest tenth of a</p><p>metre.</p><p>b) When does the ball reach its</p><p>maximum height?</p>
<ol> <li>Construction A square building of side <code class='latex inline'>\displaystyle x </code> metres is extended by <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> on one side and <code class='latex inline'>\displaystyle 5 \mathrm{~m} </code> on the other side to form a rectangle. a) Express the new area as the product of 2 binomials. b) Evaluate the new area for <code class='latex inline'>\displaystyle x=20 . </code></li> </ol>
<p>Fred wants to install a wooden deck around his rectangular swimming pool. The function</p><p><code class='latex inline'>\displaystyle C(x)=120 x^{2}+1800 x+5400 </code> represents the cost of installation, where <code class='latex inline'>\displaystyle x </code> is the width of the deck in metres and <code class='latex inline'>\displaystyle C(x) </code> is the cost in dollars. What will the width be if Fred spends <code class='latex inline'>\displaystyle \$ 9480 </code> for the deck? Here is Steve&#39;s solution.</p><p><code class='latex inline'>\displaystyle [0 </code></p><p>I used a graphing calculator to solve this problem. I entered <code class='latex inline'>\displaystyle 120 x^{2}+1800 x+5400 </code> into <code class='latex inline'>\displaystyle Y 1 </code> and 9480 into <code class='latex inline'>\displaystyle Y 2 </code> to see where they intersect. They intersect at two places: <code class='latex inline'>\displaystyle x=2 </code> and <code class='latex inline'>\displaystyle x=-17 </code>. Since both answers must be positive, use <code class='latex inline'>\displaystyle x=2 </code> and <code class='latex inline'>\displaystyle x=17 </code>. Because you will get more deck with a higher number, use only <code class='latex inline'>\displaystyle x=17 </code>.</p><p>Do you agree with his reasoning? Why or</p><p>why not?</p>
<p>A basketball is tossed from the top of a 3-m wall. The path of the basketball is defined by the relation <code class='latex inline'>y=-x^2+2x+3</code>, where <code class='latex inline'>x</code> represents the horizontal distance travelled, in metres, and <code class='latex inline'>y</code> represents the height, in metres, above the ground. How far has the basketball travelled horizontally when it lands on the ground?</p>
<p>a) Determine an equation for the perimeter of any rectangle whose width is 8 cm less than its length.</p><p>b) Determine the length of the rectangle whose width is 72 cm.</p>
<p>\star 12. The arch of a domed sports arena is in the shape of a parabola. The arch spans a width of <code class='latex inline'>\displaystyle 32 \mathrm{~m} </code> from one side of the arena to the other. The height of the arch is</p><p><code class='latex inline'>\displaystyle 18 \mathrm{~m} </code> at a horizontal distance of <code class='latex inline'>\displaystyle 8 \mathrm{~m} </code> from each end of the arch.</p><p>a) Sketch the quadratic function so that the vertex of the parabola is on the</p><p><code class='latex inline'>\displaystyle y </code>-axis and the width is along the <code class='latex inline'>\displaystyle x </code>-axis.</p><p>b) Use this information to determine</p><p>the equation that models the arch.</p><p>c) Find the maximum height of the</p><p>arch.</p>
<ol> <li>Measurement The diagonal of a rectangle is <code class='latex inline'>\displaystyle 8 \mathrm{~cm} </code>. The length of the rectangle is <code class='latex inline'>\displaystyle 1.6 \mathrm{~cm} </code> more than the width. Find the dimensions of the rectangle.</li> </ol>
<ol> <li>Measurement The hypotenuse of a right triangle measures <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code>. One of the other two sides is <code class='latex inline'>\displaystyle 2 \mathrm{~m} </code> longer than the third side. Find the unknown side lengths.</li> </ol>
<ol> <li>Framing a photograph A photograph measuring <code class='latex inline'>\displaystyle 12 \mathrm{~cm} </code> by <code class='latex inline'>\displaystyle 8 \mathrm{~cm} </code> is to be surrounded by a mat before framing. The width of the mat is to be the same on all sides of the photograph. The area of the mat is to equal the area of the photograph. Find the width of the mat.</li> </ol>
<p>Boat Building Boat builders share an old rule of thumb for sailboats. The maximum speed <code class='latex inline'>\displaystyle K </code> in knots is <code class='latex inline'>\displaystyle 1.35 </code> times the square root of the length <code class='latex inline'>\displaystyle L </code> in feet of the boat&#39;s waterline.</p><p>a. A customer is planning to order a sailboat with a maximum speed of 12 knots. How long should the waterline be?</p><p>b. How much longer would the waterline have to be to achieve a maximum speed of 15 knots?</p>
<p>The revenue, <em>R</em>, from T-shirt sales at a monthly fundraising charity event is calculated as (number of T-shirts sold) x (price of T-shirt). The current price of a T-shirt is $16, and the charity event typically sells 50 T-shirts. For each $2 increase in the price of a T-shirt, five fewer T-shirts are sold. So, the revenue can be modelled using the equation <code class='latex inline'> R = (50-5x)(16 +2x)</code>, where <em>x</em> represents the number of price increases. </p><p>a) Rewrite the equation in the form <code class='latex inline'> R = a(x-r)(x-s)</code>. </p><p>b) Sketch a graph of the relation. </p><p>c) What does the <em>R</em>-intercept represent? What do the <em>x</em>-intercepts represent?</p><p>d) What does a negative value of <em>x</em> represent?</p><p>e) What price maximizes the revenue?</p><p>f) What is the maximum revenue?</p>
<p>CCSS) REASONING The surface area of a three-dimensional object is the sum of the areas of the faces. If <code class='latex inline'>\displaystyle \ell </code> represents the length of the side of a cube, write a formula for the surface area of the cube.</p>
<p>An equipment storage shed has a parabolic cross section modelled by the relation <code class='latex inline'>\displaystyle h = - d^2 + 4d </code>, where h is the height, in metres, and d is the horizontal distance, in metres, from one edge of the shed.</p><p>a) How wide and how tall is the shed?</p><p>b) Sketch the graph.</p><p>c) For what values of d is the relation valid? Explain.</p>
<ol> <li>Integers Two consecutive integers are added. The square of their sum is 361 . What are the integers?</li> </ol>
<p>\star 10. The height of a football can be modelled by the function</p><p><code class='latex inline'>\displaystyle h(t)=-4.9 t^{2}+21.8 t+1.5 </code>, where <code class='latex inline'>\displaystyle t </code> is the time, in seconds, since the ball was thrown, and <code class='latex inline'>\displaystyle h </code> is the height of the ball, in metres, above the ground. Determine how</p><p>long the football will be in the air, to the nearest tenth of a second.</p>
<p>A ball is thrown vertically upward from an initial height of ho metres, with an initial velocity of v m/s, and is affected by the acceleration due to gravity, <code class='latex inline'>g</code>. The height function of the ball is given by <code class='latex inline'>\displaystyle h(t) = -\frac{1}{2}gt^2 + vt +h_0 </code>, where <code class='latex inline'>h(t)</code> is the height, in metres, and <code class='latex inline'>t</code> is the time, in seconds. Write an expression for the maximum height of the ball.</p>
<p>Adding 19 to the square of the number of provinces in South Africa gives the square of the number of provinces in Canada. How many provinces are there in South Africa?</p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><img src="/qimages/63654" />
<p>Clay shooting disks are launched from the ground into the air from a machine <code class='latex inline'>\displaystyle 12 \mathrm{~m} </code> above the ground. The height of each disk, <code class='latex inline'>\displaystyle h(t) </code>, in metres, is modelled by <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+30 t+12 </code>, where <code class='latex inline'>\displaystyle t </code> is the time in seconds since it was launched.</p><p>a) What is the maximum height the disks reach?</p><p>b) At what time do the disks hit the ground?</p><p>c) Determine the domain and range of this model.</p>
<p>An inflatable raft is dropped from hovering helicopter to boat in distress below. The height of the raft above the water, <code class='latex inline'>y</code>, in metres, is approximated by the equation <code class='latex inline'>y=500-5x^2</code>, where <code class='latex inline'>x</code> is the time in seconds since the raft was dropped.</p> <ul> <li>What is the height of the helicopter above the water?</li> </ul>
<p>The acceleration due to gravity, <code class='latex inline'>\displaystyle g </code>, is <code class='latex inline'>\displaystyle 9.8 \mathrm{~m} / \mathrm{s}^{2} </code> on Earth, <code class='latex inline'>\displaystyle 3.7 \mathrm{~m} / \mathrm{s}^{2} </code> on Mars, <code class='latex inline'>\displaystyle 10.5 \mathrm{~m} / \mathrm{s}^{2} </code> on Saturn, and <code class='latex inline'>\displaystyle 11.2 \mathrm{~m} / \mathrm{s}^{2} </code> on Neptune. The height, <code class='latex inline'>\displaystyle h(t) </code>, of an object, in metres, dropped from above each surface is given by <code class='latex inline'>\displaystyle h(t)=-0.5 g t^{2}+k </code></p><p>a) Describe how the graphs will differ for an object dropped from a height of <code class='latex inline'>\displaystyle 100 \mathrm{~m} </code> on each of the four planets.</p><p>b) On which planet will the object be moving fastest when it hits the surface?</p><p>c) On which planet will it be moving slowest?</p>
<p>The area of a rectangle is given by the expression <code class='latex inline'>8x^2 + 4x</code>. Draw diagrams to show the possible rectangles, labelling the length and width of each.</p>
<p>The picture shows four circles inside a square. Each small circle has a radius <code class='latex inline'>\displaystyle r </code>. The area of the shaded region as a function is <code class='latex inline'>\displaystyle A(r)=(16-4 \pi) r^{2} </code>.</p><p>a) What is the domain of the function?</p><p>b) What is the range of the function?</p><img src="/qimages/77911" />
<p>Write and simplify an expression to represent the area of each figure. </p><img src="/qimages/63490" />
<p>The manager of a hardware store sells batteries for $<code class='latex inline'>5</code> a package. She wants to see how much money she will earn if she increases the price in 10¢ increments. A model of the price change is the revenue function <code class='latex inline'>R(x)=-x^2+10x+3000</code>, where <code class='latex inline'>x</code> is the number of 10¢ increments and <code class='latex inline'>R(x)</code> is in dollars. Explain how to determine the maximum revenue.</p>
<p>A lifeguard wants to rope off a rectangular area for swimmers to swim in. She has 700 m of rope. The area, <code class='latex inline'>A(x)</code>, that is to be enclosed can be modelled by the function <code class='latex inline'>A(x) = 700 x - 2x^2</code>, where x is the width of the rectangle. What is the maximum area that can be enclosed?</p>
<p>A parabolic bridge is <code class='latex inline'>40</code>m wide. Determine the height of the bridge <code class='latex inline'>12</code> m in from the outside edge, if the height <code class='latex inline'>5</code> m in from the outside edge is <code class='latex inline'>8</code> m.</p>
<p>A helicopter drops an aid package. The height of the package above the ground at any time is modelled by the function <code class='latex inline'>h(t) = -5t^2 -30t + 675</code>, where <code class='latex inline'>h(t)</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds. How long will it take the package to hit the ground?</p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><img src="/qimages/63658" />
<p>Water from a hose is sprayed on a fire burning at a height of <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> up the side of a wall. If the function <code class='latex inline'>\displaystyle h(x)=-0.15 x^{2}+3 x </code>, where <code class='latex inline'>\displaystyle x </code> is the horizontal distance from the fire, in metres, models the height of the water, <code class='latex inline'>\displaystyle h(x) </code>, also in metres, how far back does the firefighter have to stand in order to put out the fire?</p>
<p>While hiking along the top of a cliff, Harlan knocked a pebble over the edge. The height, <code class='latex inline'>h</code>, in metres, of the pebble above the ground after <code class='latex inline'>t</code> seconds is modelled by <code class='latex inline'>h = -5t^2 - 4t + 120</code>.</p> <ul> <li>For how long is the height of the pebble greater than <code class='latex inline'>95 m</code>?</li> </ul>
<p>Roy owns hardware store. For every increase of <code class='latex inline'>10</code> cents in the price of a package of batteries, he estimates that sales decree by <code class='latex inline'>10</code> pages per day. The store normally sells 700 packages of batteries per day, at <code class='latex inline'>\$5.00</code> per package.</p><p><strong>a)</strong> Determine an equation for the revenue, <code class='latex inline'>y</code>, when <code class='latex inline'>x</code> packages of batteries are sold.</p><p><strong>b)</strong> What is the maximum daily revenue that Roy can expect from battery sales?</p><p><strong>c)</strong> How many packages of batteries are sold when the revenue is at a maximum?</p>
<ol> <li>Peace Tower a) Find the width, in metres, of the Canadian flag on the Peace Tower in Ottawa by solving the equation <code class='latex inline'>\displaystyle 8 w^{2}+18 w-81=0 </code>. b) The height of the Peace Tower is <code class='latex inline'>\displaystyle 90 \mathrm{~m} </code>. If an object is thrown downward from this height at <code class='latex inline'>\displaystyle 5 \mathrm{~m} / \mathrm{s} </code>, the approximate time, <code class='latex inline'>\displaystyle t </code> seconds, the object takes to reach the ground can be found by solving the equation <code class='latex inline'>\displaystyle -5 t^{2}-5 t+90=0 </code>. Find the time taken, to the nearest tenth of a second.</li> </ol>
<p>The entrance to a garden is an arch that can be approximated by the relation <code class='latex inline'> y = -0.2x^2 +3.2</code>, where <em>y</em> is the height, in metres, above ground and <em>x</em> is the width, in metres, from the centre of the bridge. </p><p>a) Graph the quadratic relation. </p><p>b) Describe the shape of the arch. </p><p>c) How tall and how wide is the arch?</p>
<p>A pebble is dropped from a bridge into a river. The height of the pebble above the water after it has been released is modelled by the function <code class='latex inline'>\displaystyle h(t)=80-5 t^{2} </code>, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds.</p><p>a) Graph the function for reasonable values of <code class='latex inline'>\displaystyle t </code></p><p>b) Explain why the values you chose for <code class='latex inline'>\displaystyle t </code> in part (a) are reasonable.</p><p>c) How high is the bridge? Explain.</p><p>d) How long does it take the pebble to hit the water? Explain.</p><p>e) Express the domain and range in set notation.</p>
<p>GEOMETRY The hypotenuse of a right triangle is 1 centimeter longer than one side and 4 centimeters longer than three times the other side. Find the dimensions of the triangle.</p>
<p>A baseball is hit from a height of 1 m. The height, <code class='latex inline'>h</code>, of the ball in metres after <code class='latex inline'>t</code> seconds can be modelled by <code class='latex inline'>h=-5t^2+9t+1</code>. Determine the maximum height reached by the ball.</p>
<p>A photo framer wants to place a matte of uniform width all around a photo. The area of the matte should be equal to the area of the photo. The photo measures <code class='latex inline'>40</code> cm by <code class='latex inline'>60</code> cm. How wide should the matte be?</p><img src="/qimages/2198" />
<p>Physics The function <code class='latex inline'>\displaystyle h=-16 t^{2}+1700 </code> gives an object&#39;s height <code class='latex inline'>\displaystyle h </code>, in feet, at <code class='latex inline'>\displaystyle t </code> seconds.</p><p>a. What does the constant 1700 tell you about the height of the object?</p><p>b. What does the coefficient of <code class='latex inline'>\displaystyle t^{2} </code> tell you about the direction the object is moving?</p><p>c. When will the object be 1000 ft above the ground?</p><p>d. When will the object be 940 ft above the ground?</p><p>e. What are a reasonable domain and range for the function <code class='latex inline'>\displaystyle h </code> ?</p>
<p>The path of a ball is modelled by the equation <code class='latex inline'>y=-x^2+4x+1</code>, where <code class='latex inline'>x</code> is the horizontal distance, in metres, travelled and y is the height, in metres, of the ball above the ground. What is the maximum height of the ball, and at what horizontal distance does it occur?</p>
<p>For the relation, explain what each coordinate of the vertex represents and what the zeros represent.</p> <ul> <li>a relation that models the cost, <code class='latex inline'>C</code>, to create <code class='latex inline'>n</code> items using a piece of machinery</li> </ul>
<p>The relation <code class='latex inline'>h = -4.9t^2 + 120t + 3</code> defines the height of a rocket, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds following its launch. If you want to determine how long the rocket was in flight, what must you do?</p><p>A. Determine the vertex.</p><p>B. Substitute <code class='latex inline'>t = 0</code>, and solve for <code class='latex inline'>h</code>.</p><p>C. Complete the square.</p><p>D. Determine the zeros of the relation.</p>
<p>The underside of a bridge forms a parabolic arch. The arch has a maximum height of 30 m and a width of 50. Can a sailboat pass under the bridge, 8 m from the axis of symmetry, if the top of its mast is 27 m above the water? Justify your solution.</p>
<p>In an electrical circuit, the voltage, <code class='latex inline'>V</code> volts, as a function of time, <code class='latex inline'>t</code> minutes, is modelled by the quadratic function <code class='latex inline'>V(t) = 2t^2 - 9t + 12</code>.</p><p>a) Determine the minimum and maximum voltages during the first 5 min.</p><p>b) At what times do the values found in part a) occur?</p>
<p>The hypotenuse of a right triangle is 17 cm long. Another side of the triangle is 7 cm longer than the third side. Determine the unknown side lengths.</p>
<p>A parabola has equation <code class='latex inline'> y = 3(x+2)^2 + 4</code>. Write an equation for the parabola after each set of transformations. </p><p>a) a reflection in the <em>x</em>-axis. </p><p>b) a translation 6 units to the right. </p><p>c) a reflection in the <em>x</em>-axis, followed by a translation of 3 units downward. </p><p>d) a reflection in the <em>y</em>-axis. </p>
<p>The quadratic function <code class='latex inline'>d(s)=0.0056s^2+0.14s</code> models the relationship between stopping distance, <code class='latex inline'>d</code>, in metres and speed, <code class='latex inline'>s</code>, in kilometres per hour in driving a car. What is the fastest you can drive and still be able to stop within <code class='latex inline'>60</code> m?</p>
<p>The perimeter of a right triangle is 36.0 cm and the length of the hypotenuse is 15.0 cm. Determine the length of the other two sides</p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><img src="/qimages/63653" />
<p>The height of a soccer ball kicked in the air is given by the quadratic equation <code class='latex inline'>h(t)=-4.9t(t-2.1)^2+23</code>, where time, <code class='latex inline'>t</code>, is in seconds and height, <code class='latex inline'>h(t)</code>, is in metres.</p><p> What is the maximum height of the ball?</p>
<p>The daily revenue, <code class='latex inline'>\displaystyle R </code>, of a small clothing boutique depends on the price, <code class='latex inline'>\displaystyle d </code>, at which each dress is sold. <code class='latex inline'>\displaystyle R=-2(d-135)^{2}+1500 </code> models the daily revenue. The owner of the boutique has discovered that her maximum daily revenue will increase by <code class='latex inline'>\displaystyle \$ 250 </code> if she increases the price of each dress by <code class='latex inline'>\displaystyle \$ 28 . </code> What will be the new daily revenue model?</p><p>A. <code class='latex inline'>\displaystyle R=-2(d-135)^{2}+1528 </code></p><p>B. <code class='latex inline'>\displaystyle R=-2(d-385)^{2}+1500 </code></p><p>C. <code class='latex inline'>\displaystyle R=-2(d-28)^{2}+250 </code></p><p>D. <code class='latex inline'>\displaystyle R=-2(d-163)^{2}+1750 </code></p>
<p>A baseball is tossed into the air and follows a path <code class='latex inline'>h = -2t^2 + 6t</code>, where <em>t</em> is the time, in seconds, and <em>h</em> is the height of a baseball, in metres. </p><p>a) Find the maximum height of the baseball. </p><p>b) At what time will the baseball reach its maximum height?</p>
<p>The function <code class='latex inline'>\displaystyle h(t)=1+4 t-1.86 t^{2} </code> models the height of a rock thrown upward on the planet Mars, where <code class='latex inline'>\displaystyle h(t) </code> is height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. Use a graph to determine a) the maximum height the rock reaches</p><p>b) how long the rock will be above the surface of Mars</p>
<p>If a baseball is batted at an angle of 35° to the ground, the distance the ball travels can be estimated using the equation <code class='latex inline'>d = 0.0034s^2 + 0.004s - 0.3</code>, where sis the bat speed, in kilometres per hour, and <code class='latex inline'>d</code> is the distance flown. in metres. At what speed does the batter need to hit the ball in order to have a home run where the ball flies <code class='latex inline'>125</code> m? Round to the nearest tenth.</p>
<p>The population of a city, <code class='latex inline'>\displaystyle P(t) </code>, is modelled by the quadratic function <code class='latex inline'>\displaystyle P(t)=50 t^{2}+1000 t+20000 </code>, where <code class='latex inline'>\displaystyle t </code> is time in years. Note:</p><p><code class='latex inline'>\displaystyle t=0 </code> corresponds to the year 2000 . Peg says that the population was 35000 in 1970 . Explain her reasoning for choosing that year.</p>
<p>Each expression represents the side length of a cube. Write a polynomial in standard form for the surface area of each cube.</p><p><code class='latex inline'>\displaystyle 2 c^{2}+3 </code></p>
<p>The height in metres of projectile is modelled by function <code class='latex inline'>h(t) = -5t^2 + 24</code>, where <code class='latex inline'>t</code> is the time seconds.</p><p>a) Find the point when the object hits the ground.</p><p>b) Find the average rate of change from the point when the projectile is launched (<code class='latex inline'>t=0</code>) to the point which it hits the ground.</p><p>c) Estimate the object&#39;s speed at the point of impact.</p>
<p>Meg went bungee jumping from the Bloukrans River bridge in South Africa last summer. During the free fall on her first jump, her height above the water, <code class='latex inline'>h</code>, in metres, was modelled by <code class='latex inline'>h=-5t^2+t+216</code>, where <code class='latex inline'>t</code> is the time in seconds since she jumped.</p> <ul> <li> How high above the water is the platform from which she jumped?</li> </ul>
<p>The UV index on a sunny day can be modelled by the function <code class='latex inline'>f(x) =-0.15(x - 13)^2+ 7.6</code>, where <code class='latex inline'>x</code> represents the time of day on a 24-h clock and <code class='latex inline'>f(x)</code> represents the UV index. Between what hours was the UV index greater than <code class='latex inline'>7</code>?</p>
<p>The platforms on the ends of the half-pipe are at the same height. </p><img src="/qimages/1118" /><p>a) How wide is the half-pipe?</p>
<img src="/qimages/63942" /><p> Golden Gate Bridge The road on the Golden Gate Bridge is supported by two towers and the two cables that join them. The distance between the towers is <code class='latex inline'>\displaystyle 1280 \mathrm{~m} </code>. Suppose the curve of a cable is graphed on a grid, with the origin on the road at the centre of the bridge. The curve made by the cable is a catenary that can be approximately modelled by the quadratic function</p><p><code class='latex inline'>\displaystyle h=0.00037 d^{2}+2 </code></p><p>where <code class='latex inline'>\displaystyle h </code> metres is the height of the cable above the road, and <code class='latex inline'>\displaystyle d </code> metres is the horizontal distance from the centre of the bridge. a) Graph the function.</p><p>b) What is the distance from the road to the lowest point of the cable? c) What is the maximum height of the towers above the road, to the nearest ten metres?</p><p>d) At a horizontal distance of <code class='latex inline'>\displaystyle 200 \mathrm{~m} </code> from the centre of the bridge, how high is the cable above the road, to the nearest metre?</p>
<p>A T-ball player hits a ball from a tee that is 0.6 m tall. The height of the ball at a given time is modelled by the function <code class='latex inline'>h(t) = -4.9t^2 + 7t + 0.6</code>, where height, h(t), is in metres and time, <code class='latex inline'>t</code>, is in seconds. </p><p>a) What will the height be after 1s?</p><p>b) When will the ball hit the ground?</p>
<p>In a volleyball match, Jenny serves the volleyball at 14 m/s, from a height of 2.5 m above the court. The height of the ball in flight can be estimated using the equation <code class='latex inline'>\displaystyle h = -4.9t^2+14t+2.5 </code>, where <code class='latex inline'>h</code> is the height, in metres, and t is the time, in seconds, after she serves the ball.</p><p>a) What is the maximum height of the volleyball above the court? When does this occur? Round answer to the nearest to the nearest tenth.</p><p>b) If a player on the other team contacts the ball at a height of 0.5 In above the court, how long does it take for the ball to reach her? Round to the nearest second.</p>
<p>On Mars, if you hit a baseball, the height of the ball at time <code class='latex inline'>t</code> would be modelled by the quadratic function <code class='latex inline'>h(t)=-1.85t^2+20t+1</code>, where <code class='latex inline'>t</code> is in seconds and <code class='latex inline'>h(t)</code> is in metres.</p><p>When will the ball hit the ground?</p>
<p>The traffic safety bureau receives data regarding acceleration of a prototype electric sports car. It can accelerate from 0 to 100 km/h in about 4 s. Its position, d, in metres, at any time t, in seconds, is given by <code class='latex inline'>d(t)=3.5t^2</code>. Mathew is comparing the prototype to a hybrid electric car, which has its position given by <code class='latex inline'>d(t)=1.4t^2</code>. </p><p><strong>(a)</strong> In a race between the two cars, the hybrid is given a head start. Where would the hybrid have to start so that after 4 s of acceleration, both cars are in the same position?</p>
<p>A toy rocket is launched from a <code class='latex inline'>3</code>-m platform, at <code class='latex inline'>8.1</code> m/s. The height of the rocket is modelled by the equation <code class='latex inline'>h=-4.9t^2+8.1t+3</code>,where <code class='latex inline'>h</code> is the height, in metres, above the ground and <code class='latex inline'>t</code> is the time, in seconds.</p><p>a) After how many seconds will the rocket rise to a height of 6 m above the ground? Round your answer to the nearest hundredth.</p><p>b) When does the rocket fall again to a height of 6 m above the ground? Use your answers from parts a] and b) to determine when the rocket reached its maximum height above the ground.</p><p><code class='latex inline'>\to</code> c) Use your answers from parts a] and b) to determine when the rocket reached its maximum height above the ground.</p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><img src="/qimages/63657" />
<p>The function <code class='latex inline'>A(w) = 576w -2w^2</code> models the area of a pasture enclosed by a rectangular fence, where <code class='latex inline'>w</code> is width in metres.</p><p>a) What is the maximum area that can be enclosed?</p><p>b) Determine the area that can be enclosed using a width of 20 m.</p><p>c) Determine the width of the rectangular pasture that has an area of <code class='latex inline'>18 144 m^2</code>.</p>
<p>The quadratic relation <code class='latex inline'>\displaystyle h=-5 t^{2}+80 t </code> models the height, <code class='latex inline'>\displaystyle h </code>, in metres, that an object projected upward from the ground will reach in <code class='latex inline'>\displaystyle t </code> seconds following its launch. What is the maximum height that this object will reach?</p><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { A. } 80 \mathrm{~m} & \text { C. } 320 \mathrm{~m} \\ \text { B. } 400 \mathrm{~m} & \text { D. } 100 \mathrm{~m}\end{array} </code></p>
<p>The height, l), in metres, of a baseball after Bill hits it with a hat is described by the function <code class='latex inline'>h(t) = 0.8 +29.4t -4.9t^2</code>, where tis the time in seconds after the ball is struck. What is the maximum height of the hall?</p><p>A. 4.9 m</p><p>B. 29.4</p><p>C. 44.9 m</p><p>D. 25 m</p>
<p>The UV index on a sunny day can be modelled by the function <code class='latex inline'>\displaystyle f(x) = -0.15(x - 12.5)^2 + 8.6 </code> where x represents the tlme of day on a 24—h clock and <code class='latex inline'>f(x)</code> represents the UV index. Between what hours was the UV index more than 8?</p>
<p>The grass in the backyard of a house is a square with side length <code class='latex inline'>10</code> m. A square patio is placed in the centre. If the side length, in metres, of the patio is <code class='latex inline'>x</code>, then the area of grass remaining is given by the relation <code class='latex inline'>A = -x^2 + 100</code>.</p><img src="/qimages/1814" /> <ul> <li>Find the intercepts. What do they represent?</li> </ul>
<p>A football is kicked into the air. Its height above the ground is approximated by the relation <code class='latex inline'>h=20t-5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds since the football was kicked.</p> <ul> <li>What is the maximum height reached by the football? After how many seconds does the maximum height occur?</li> </ul>
<ol> <li>Integers The sum of the squares of three consecutive positive integers is <code class='latex inline'>\displaystyle 194 . </code> Find the integers.</li> </ol>
<p>A company that manufactures MP3 players uses the relation <code class='latex inline'>P=120x-60x^2</code> to model its profit. The variable <code class='latex inline'>x</code> represents the number of thousands of MP3 players sold. The variable <code class='latex inline'>P</code> represents the profit in thousands of dollars.</p> <ul> <li>What is the maximum profit the company can earn?</li> </ul>
<p>The triangle with sides given by <code class='latex inline'>x^2 + 1, x^2 -1</code> and <code class='latex inline'>2x</code> will always be right triangle for <code class='latex inline'>x > 1</code>.</p><p>Based on the three expressions four sides, which one must represent the hypotenuse? Justify your answer.</p>
<p>A toy rocket is launched from a 3-m platform, at <code class='latex inline'>8.1</code> m/s. The height of the rocket is modelled by the equation <code class='latex inline'>h=-4.9t^2+8.1t+3</code>,where <code class='latex inline'>h</code> is the height, in metres, above the ground and <code class='latex inline'>t</code> is the time, in seconds.</p><p>a) After how many seconds will the rocket rise to a height of 6 m above the ground? Round your answer to the nearest hundredth.</p><p>b) When does the rocket fall again to a height of 6 m above the ground? Use your answers from parts a] and b) to determine when the rocket reached its maximum height above the ground.</p>
<p>Conor has a summer lawn-mowing business. Based on experience, Conor knows that <code class='latex inline'>P=-5x^2+200x-1500</code> models his profit, <code class='latex inline'>P</code>, in dollars, where <code class='latex inline'>x</code> is the amount, in dollars, charged per lawn.</p> <ul> <li>How much does he need to charge if he wants to break even?</li> </ul>
<ol> <li>Integers The sum of the squares of two consecutive even integers is 452 . Find the integers.</li> </ol>
<p>The sum of the squares of three consecutive integers is 194. Determine the integers.</p>
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p> <ul> <li></li> </ul>
<p>Sean and Farah have 24 m of fencing to enclose a vegetable garden at the back of their house. Determine the dimensions of the largest rectangular garden they could enclose, using the back of their house as one of the sides of the rectangle.</p>
<p>You are designing a chute for loading grain into rail cars. The cars have a round hatch 60 cm in diameter. The chute will have a square cross section. Find the side length of the largest chute that will fit into the hatch. Round your answer to the nearest centimetre.</p><img src="/qimages/7092" />
<p>Alysia has selected the letter E to design the logo for her school team, the Eagles.</p><p>The design will be used to make different-sized crests for clothing such as jackets, sweaters, and baseball caps. The height of the crest is twice the width. How can Alysia make sure that, when the crest is made larger or smaller, the proportions will not change?</p><p>a) Find an expression for the area of the crest in terms of the width.</p><p>b) Determine the area of a crest with a width of 8 cm.</p><p>c) Determine the height of a crest with an area of <code class='latex inline'>72 cm^2</code>.</p>
<p>Stacey maintains the gardens in the city parks. In the summer, she plans to build a walkway through the rose garden. The area of the walkway, <code class='latex inline'>A</code>, in square metres, is given by <code class='latex inline'>A=160x+4x^2</code>, where <code class='latex inline'>x</code> is the width of the walkway in metres. If the area of the walkway must be 900 m<code class='latex inline'>^2</code>, determine the width.</p>
<ol> <li>Skating rink A rectangular skating rink measures <code class='latex inline'>\displaystyle 40 \mathrm{~m} </code> by <code class='latex inline'>\displaystyle 20 \mathrm{~m} </code>. It is to be doubled in area by extending each side by the same amount. Determine how much each side should be extended, to the nearest tenth of a metre.</li> </ol>
<p>a) Is the relationship in the graph linear or non-linear. Explain.</p><img src="/qimages/22047" /><p>b) How many x-intercepts does the graph have? What are they?</p><p>c) How many y—intercepts does the graph have? What are they?</p><p>d) Sketch the graph of a relation that has the same shape as the given relation with one x—intercept and one y-intercept.</p><p>e) Sketch the graph of a relation that has the same shape as the given relation with no x—intercept and one y-intercept.</p>
<p>The product of two consecutive integers is 1482. What are the numbers?</p>
<p>The population, <code class='latex inline'>\displaystyle P(t) </code>, of an Ontario city is modelled by the function <code class='latex inline'>\displaystyle P(t)=14 t^{2}+650 t+32000 . </code> Note: <code class='latex inline'>\displaystyle t=0 </code> corresponds to the year <code class='latex inline'>\displaystyle 2000 . </code></p><p>a) What will the population be in 2035 ? b) When will the population reach 50000 ? c) When was the population 25000 ?</p>
<ol> <li>Numbers The product of two consecutive even numbers is 288 . What are the numbers?</li> </ol>
<p>A ball is thrown vertically upward from the top of a cliff. The height of the ball is modelled by the function <code class='latex inline'>\displaystyle h(t)=65+10 t-5 t^{2} </code>, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. Determine when the ball reaches its maximum height.</p>
<p>A soccer ball is kicked into the air and follows the path <code class='latex inline'> h = -2t^2 + 12r</code>, where <em>t</em> is the time, in seconds, and <em>h</em> is the height of the soccer ball, in metres. </p><p>a) Find the maximum height of the soccer ball. </p><p>b) At what time will the soccer ball reach its maximum height?</p><p>c) How long will the soccer ball be in the air?</p><p>d) How long does it take the soccer ball to reach a height of 16 m?</p>
<ol> <li>Volleyball The area of a volleyball court, excluding the service areas, can be represented by the trinomial <code class='latex inline'>\displaystyle 2 x^{2}-4 x+2 </code>. a) Factor the trinomial completely.</li> </ol> <p>b) If the length of the court is twice the width, use the factors from part a) to write expressions that represent the length and the width. c) If <code class='latex inline'>\displaystyle x </code> represents <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code>, what are the length and the width of the court, in metres?</p>
<p>The path of a ball is modelled by the quadratic function <code class='latex inline'>h(t) = -5(t - 2)^2 + 23</code> , where height, <code class='latex inline'>h(t)</code>, is in metres and time, <code class='latex inline'>t</code>, is in seconds.</p><p>a) What is the maximum height the ball reaches?</p><p>b) When does it reach the maximum height?</p><p>c) When will the ball reach a height of 18 m?</p>
<p>A volleyball’s height, <code class='latex inline'>h</code>, in metres, above the ground after 1 seconds is modelled by the relation <code class='latex inline'>h = -4.9t^2 + 5t + 2</code>.</p><p>a) Graph the relation.</p><p>b) What is the h-intercept? What does it represent?</p><p>c) How long will it take the volleyball to hit the ground? What feature on the graph models this? Explain your answer.</p>
<p>A rectangular field has a perimeter of <code class='latex inline'>500m</code> and an area of <code class='latex inline'>14400 m^2</code>. Determine the dimensions of the field.</p>
<ol> <li>Measurement A cylinder has a height of <code class='latex inline'>\displaystyle 5 \mathrm{~cm} </code> and a surface area of <code class='latex inline'>\displaystyle 100 \mathrm{~cm}^{2} </code>. Find the radius of the cylinder, to the nearest tenth of a centimetre.</li> </ol>
<p>Eighteen more than the square of an integer is 43. What is the integer?</p>
<ol> <li>Measurement The width of a rectangle is <code class='latex inline'>\displaystyle 1 \mathrm{~m} </code> less than the length. The area is <code class='latex inline'>\displaystyle 72 \mathrm{~m}^{2} </code>. Find the width and the length.</li> </ol>
<p>A company that makes modular furniture has designed a scalable box to accommodate several different sizes of items. The dimensions are given by <code class='latex inline'>L =2x + 0.5, W = x -0.5</code>, and <code class='latex inline'>H = x + 0.5</code>, where <code class='latex inline'>x</code> is in metres. </p><p>State the domain and range of the volume and surface area functions.</p>
<ol> <li>The function <code class='latex inline'>\displaystyle d=0.0067 v^{2}+0.15 v </code> can be used to determine the safe stopping</li> </ol> <p>distance, <code class='latex inline'>\displaystyle d </code>, in metres, for a car given its speed, <code class='latex inline'>\displaystyle v </code>, in kilometres per hour. Determine the speed at which a car can be travelling in order to be able to stop in the given distances.</p><p>a) <code class='latex inline'>\displaystyle 24 \mathrm{~m} </code></p><p>b) <code class='latex inline'>\displaystyle 43 \mathrm{~m} </code></p><p>c) <code class='latex inline'>\displaystyle 82 \mathrm{~m} </code></p>
<p>April sells specialty teddy bears at various summer festivals. Her profit for a week, <code class='latex inline'>P</code>, in dollars, can be modelled by <code class='latex inline'>P = -0.1n^2 + 30n - 1200</code>, where <code class='latex inline'>n</code> is the number of teddy bears she sells during the week.</p><p>a) According to this model, could April earn a profit of in one week? Explain.</p><p>b) How many teddy bears would she have to sell to break even?</p><p>c) How many teddy bears would she have to sell to earn $500?</p><p>d) How many teddy bears would she have to sell to maximize her profit?</p>
<p>A soccer ball is kicked into the air. lts height, I}, in metres, is approximated by the equation <code class='latex inline'>h = -5t^2 + 15t + 0.5</code>, where <code class='latex inline'>t</code> is the time in seconds since the ball was kicked.</p><p>a) From what height is the ball kicked? </p><p>b) When does the ball hit the ground?</p><p>c) When does the ball reach its maximum height?</p><p>d) What is the maximum height of the ball?</p><p>e) What is the height of the ball at <code class='latex inline'>t = 3</code>? Is the ball travelling upward or downward at this time? Explain.</p><p>f) When is the ball at a height of 10 m?</p>
<p>The expression <code class='latex inline'>16t^2</code> models the distance in feet that an object falls during the first <code class='latex inline'>t</code> seconds after being dropped. What is the distance the object falls during each time?</p><p>2 second</p>
<p>A bridge is going to be constructed over a river. The underside of the a bridge will form a parabolic arch, as shown in the picture. The river is 18 m wide and the arch will be anchored on the ground, 3 m back from the riverbank on both sides. The maximum height of the arch must be between 22 m and 26 m above the surface of the river. Create two different equations to represent arches that satisfy these conditions. Then use graphing technology to graph your equations on the same grid.</p><img src="/qimages/1538" />
<p>It is estimated that, <code class='latex inline'>t</code> years from now, that amount of waste accumulated <code class='latex inline'>Q</code>, in tonnes, will be <code class='latex inline'>Q(t)=10^4(t^2+15t+70), 0\leq t \leq 10</code>.</p><p>How much waste has been accumulated up to now?</p>
<ol> <li>Measurement Find the dimensions of both squares in each diagram. The area of the shaded region is given in each case.</li> </ol> <img src="/qimages/64501" />
<p>The cost, <code class='latex inline'>\displaystyle C(n) </code>, in dollars, of operating a concrete-cutting machine is modelled by <code class='latex inline'>\displaystyle C(n)=2.2 n^{2}-66 n+655 </code>, where <code class='latex inline'>\displaystyle n </code> is the number of minutes the machine is in use.</p><p>a) How long must the machine be in use for the operating cost to be a minimum?</p><p>b) What is the minimum cost?</p>
<p>The stopping distance for a boat in calm water is modelled by the function <code class='latex inline'>d(v)=0.004v^2+0.2v+6</code>, where <code class='latex inline'>d(v)</code> is in metres and <code class='latex inline'>v</code> is in kilometres per hour.</p><p>What is the stopping distance if the speed is <code class='latex inline'>10</code> km/h?</p>
<p>A triangle has base <code class='latex inline'>2x + 1</code> and height <code class='latex inline'>6x - 3</code>. What value of <code class='latex inline'>x</code> would give an area of <code class='latex inline'>240 m^2</code>? Round to the nearest hundredth.</p>
<p>An inflatable raft is dropped from hovering helicopter to boat in distress below. The height of the raft above the water, <code class='latex inline'>y</code>, in metres, is approximated by the equation <code class='latex inline'>y=500-5x^2</code>, where <code class='latex inline'>x</code> is the time in seconds since the raft was dropped.</p> <ul> <li>When does the raft reach the water?</li> </ul>
<p>Think About a Plan Suppose you want to put a frame around the painting shown at the right. The frame will be the same width around the entire painting. You have 276 in. <code class='latex inline'>\displaystyle ^{2} </code> of framing material. How wide should the frame be?</p> <ul> <li><p>What does 276 in. <code class='latex inline'>\displaystyle ^{2} </code> represent in this situation?</p></li> <li><p>How can you write the dimensions of the frame using two binomials?</p></li> </ul> <img src="/qimages/89993" />
<p> A rectangular field is to be enclosed and divided into two sections by a fence parallel to one of the sides using a total of 600 m of fencing. What is the maximum area that can be enclosed and what dimensions will give this area?</p>
<p>The approximate cost of operating a certain car at a constant speed is given by the formula <code class='latex inline'>C= 0.006(s-50)^2 + 20</code>, for <code class='latex inline'>10 \leq s \leq 130</code>, where <code class='latex inline'>s</code> is the speed, in kilometres per hour, and <code class='latex inline'>C</code> is the most, in cents per kilometre. Use a graphing calculator to compare the operating costs, at different speeds, to those of a second vehicle with formula <code class='latex inline'>C = 0.008(s-55)^2+15</code></p>
<p>A ball is thrown upward from the</p><p>balcony of an apartment building and</p><p>falls to the ground. The height of the</p><p>ball, <code class='latex inline'>\displaystyle h </code> metres, above the ground after <code class='latex inline'>\displaystyle t </code> seconds is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+15 t+55 . </code> a) Determine the maximum height of</p><p>the ball.</p><p>b) How long does it take the ball to</p><p>reach its maximum height?</p><p>c) How high is the balcony?</p>
<p>The path of a shot put is given by</p><p><code class='latex inline'>\displaystyle h(d)=0.0502\left(d^{2}-20.7 d-26.28\right) </code>, where <code class='latex inline'>\displaystyle h(d) </code> is the height and <code class='latex inline'>\displaystyle d </code> is the horizontal distance, both in metres. a) Rewrite the relation in the form <code class='latex inline'>\displaystyle h(d)=a(d-r)(d-s) </code>, where <code class='latex inline'>\displaystyle r </code> and <code class='latex inline'>\displaystyle s </code> are the zeros of the relation. b) What is the significance of <code class='latex inline'>\displaystyle r </code> and <code class='latex inline'>\displaystyle s </code> in this question?</p>
<p> Communication a) Describe the pattern in words.</p><p>b) Copy and evaluate the numerical expressions.</p><p><code class='latex inline'>\displaystyle (2 \times 3)-(1 \times 4) </code></p><p>c) Generalize the pattern by letting the first number in the second</p><p><code class='latex inline'>\displaystyle (3 \times 4)-(2 \times 5) </code> bracket be <code class='latex inline'>\displaystyle x </code>. Write expressions for the other numbers in terms of <code class='latex inline'>\displaystyle x </code>.</p><p><code class='latex inline'>\displaystyle (4 \times 5)-(3 \times 6) </code> Then, write an algebraic expression that matches the pattern in the numerical expressions. Expand and simplify the algebraic expression.</p><p><code class='latex inline'>\displaystyle (5 \times 6)-(4 \times 7) </code></p><p>d) How does the simplified expression explain the answer to part b)?</p>
<p>Board-feet are used to measure the total length, in feet, of boards that are <code class='latex inline'>1</code> inch thick and <code class='latex inline'>1</code> foot wide that can be cut from a tree to make lumber. You can use the equation <code class='latex inline'>l=0.011a^2-0.68a+13.31</code> </p><p><strong>unfinished question</strong></p>
<p>The student council is organizing a trip to a rock concert. All proceeds from ticket sales will be donated to charity. Tickets to the concert cost <code class='latex inline'>\displaystyle \$ 31.25 </code> per person if a minimum of 104 people attend. For every 8 extra people that attend, the price will decrease by <code class='latex inline'>\displaystyle \$ 1.25 </code> per person. a) How many tickets need to be sold to</p><p>maximize the donation to charity?</p><p>b) What is the price of each ticket that maximizes the donation?</p><p>c) What is the maximum donation?</p>
<p>Maria produces and sells shell necklaces. The material for each necklace costs her $4. She has been selling them for $8 each and averaging sales of 40 per week. She has been told that she could charge more but has found that for each $0.50 increase in price, she would lose 4 sales each week. What selling price should she set and what would her profit per week be at this price?</p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><img src="/qimages/63656" />
<p>Write an expression for the area of each shape. Expand and simplify. </p><img src="/qimages/2215" />
<ol> <li>Geometry For triangles in which the base and the height are equal, a) write an equation that relates the area, <code class='latex inline'>\displaystyle A </code>, to the height, <code class='latex inline'>\displaystyle h </code> b) graph <code class='latex inline'>\displaystyle A </code> versus <code class='latex inline'>\displaystyle h </code> c) find the <code class='latex inline'>\displaystyle h </code> - and <code class='latex inline'>\displaystyle A </code> -intercepts d) state the domain and range</li> </ol>
<p>. Measurement The area of the rectangle shown in the diagram is <code class='latex inline'>\displaystyle 36 \mathrm{~cm}^{2} </code>. What are its dimensions?</p><img src="/qimages/156996" />
<p>A baseball is thrown upward at an initial velocity of 9.2 m/s, from a height of 1.6 m above the ground. The height of the baseball, in metres, above the ground after <code class='latex inline'>t</code> seconds is modelled by the equation <code class='latex inline'>h=-4.9t^2+9.2t+1.6</code>.</p><p>a) How long does it take the baseball to fall to the ground, rounded to the nearest tenth of a second?</p><p>b) Find the times when the baseball is at a height of 4.5 m above the ground. Round your answers to the nearest tenth of a second.</p><p>c) What is the maximum height of the baseball? At what time does it reach this height? Round your answers to the nearest tenth.</p>
<p>A baseball is hit from a height of <code class='latex inline'>\displaystyle 1 \mathrm{~m} </code>. The height of the ball is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+10 t+1 </code>, where <code class='latex inline'>\displaystyle t </code> is time in seconds.</p><p>a) Graph the function for reasonable values of <code class='latex inline'>\displaystyle t </code>.</p><p>b) Explain why the values you chose for <code class='latex inline'>\displaystyle t </code> in part (a) are reasonable.</p><p>c) What is the maximum height of the ball?</p><p>d) At what time does the ball reach the maximum height?</p><p>e) For how many seconds is the ball in the air?</p><p>f) For how many seconds is the ball higher than <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> ?</p><p>g) Express the domain and range in set notation.</p>
<p>The revenue for a business is modelled by the function <code class='latex inline'>R(x) = -2,8(x -10)^2 + 15</code>, where <code class='latex inline'>x</code> is the number of items sold, in thousands, and <code class='latex inline'>R(x)</code> is the revenue in thousands of dollars. Express the number sold in terms of the revenue.</p>
<p>An inflatable raft is dropped from hovering helicopter to boat in distress below. The height of the raft above the water, <code class='latex inline'>y</code>, in metres, is approximated by the equation <code class='latex inline'>y=500-5x^2</code>, where <code class='latex inline'>x</code> is the time in seconds since the raft was dropped.</p> <ul> <li>When is the raft 100 m above the water?</li> </ul>
<p>A transport truck <code class='latex inline'>3</code> m wide and <code class='latex inline'>4</code> m tall is attempting to pass under a parabolic bridge that is <code class='latex inline'>6</code> m wide at the base and <code class='latex inline'>5</code> m high at the centre. (Ian the truck make it under the bridge? If so, how much clearance will the truck have? If not, how much more clearance is needed?</p>
<p>The area, <code class='latex inline'>A</code>, of a square is related to its perimeter, <code class='latex inline'>P</code>, by the formula <code class='latex inline'>A=\displaystyle{\frac{P^2}{16}}</code></p><p>Find the perimeter of a square with area</p> <ul> <li>i. 25 <code class='latex inline'>cm^2</code></li> <li>ii. 50 <code class='latex inline'>cm^2</code></li> </ul>
<p>Martin wants to build an additional closet in a corner of his bedroom. Because the closet will be in a corner, only two new walls need to be built. The total length of the two new walls must be <code class='latex inline'>12</code> m. Martin wants the length of the closet to be twice as long as the width, as shown in the diagram. </p><p>Let function <code class='latex inline'>f(l)</code> be the sum of the tenth and the width. Find the equation for <code class='latex inline'>f(l)</code>. Show your work.</p>
<p>An open-topped box is to be constructed from a square piece of cardboard by removing a square with side length 8 cm from each corner and folding up the edges. The resulting box is to have a volume of 512 cm3. Find the dimensions of the original piece of cardboard.</p><img src="/qimages/6030" />
<p>A rectangle has a width of <code class='latex inline'>\displaystyle 2 x-3 </code> and a length of <code class='latex inline'>\displaystyle 3 x+1 . </code> a) Write its area as a simplified polynomial. b) Write expressions for the dimensions if the width is doubled and the length is increased by <code class='latex inline'>\displaystyle 2 . </code> c) Write the new area as a simplified polynomial.</p>
<p>A field-hockey ball must stay below waist height, approximately <code class='latex inline'>\displaystyle 1 \mathrm{~m} </code>, when shot; otherwise, it is a dangerous ball. Sally hits the ball. The function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+10 t </code>, where <code class='latex inline'>\displaystyle b(t) </code> is in metres and <code class='latex inline'>\displaystyle t </code> is in seconds, models the height of the ball. Has she shot a dangerous ball? Explain.</p>
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p> <ul> <li></li> </ul>
<p>The cost, <code class='latex inline'>\displaystyle C(n) </code>, of operating a cement-mixing truck is modelled by the function</p><p><code class='latex inline'>\displaystyle C(n)=2.2 n^{2}-66 n+700 </code>, where <code class='latex inline'>\displaystyle n </code> is the number of minutes the truck is running. What is the minimum cost of operating the truck?</p>
<p>Physics The equation <code class='latex inline'>\displaystyle h=80 t-16 t^{2} </code> models the height <code class='latex inline'>\displaystyle h </code> in feet reached in <code class='latex inline'>\displaystyle t </code> seconds by an object propelled straight up from the ground at a speed of <code class='latex inline'>\displaystyle 80 \mathrm{ft} / \mathrm{s} </code>. Use the discriminant to find whether the object will ever reach a height of <code class='latex inline'>\displaystyle 90 \mathrm{ft} </code>.</p>
<ol> <li>Measurement The length and width of a rectangle are <code class='latex inline'>\displaystyle 6 \mathrm{~m} </code> and <code class='latex inline'>\displaystyle 4 \mathrm{~m} </code>. When each dimension is increased by the same amount, the area of the new rectangle is <code class='latex inline'>\displaystyle 50 \mathrm{~m}^{2} </code>. Find the dimensions of the new rectangle, to the nearest tenth of a metre.</li> </ol>
<p>The profit function for a new product is given by <code class='latex inline'>P(x) = -4x^2 + 28x -40</code>, where <code class='latex inline'>x</code> is the number sold in thousands. How many items must be sold for the company to break even?</p><p>a) 2000 or 5000</p><p>b) 2000 or 3500</p><p>c) 5000 or 7000</p><p>d) 3500 or 7000</p>
<p>A circular pizza has a radius of <code class='latex inline'>x</code> cm.</p><p>a) Write an expression for the area of the pizza.</p><p>b) Write an expression for the area of a pizza with a radius that is <code class='latex inline'>5</code> cm greater.</p><p>c) How much greater is the second area? Write the difference as a simplified expression. </p>
<p>Determine the maximum revenue generated by the manager.</p><p>The manager of a hardware store sells batteries for \$<code class='latex inline'>5</code> a package. She wants to see how much money she will earn if she increases the price in 10¢ increments. A model of the price change is the revenue function <code class='latex inline'>R(x)=-x^2+10x+3000</code>, where <code class='latex inline'>x</code> is the number of 10¢ increments and <code class='latex inline'>R(x)</code> is in dollars. Explain how to determine the maximum revenue.</p>
<p>Find two consecutive whole numbers such that the sum of their squares is 265.</p>
<p>A video tracking device recorded the height, <code class='latex inline'>h</code>, in metres, of a baseball after it was hit. The data collected can be modelled by the relation <code class='latex inline'>h=-5(t-2)^2+21</code>, where <code class='latex inline'>t</code> is the time in seconds after the ball was hit.</p> <ul> <li>Sketch a graph that represents the height of the baseball.</li> </ul>
<p>A farming cooperative has recorded information about the relationship between tonnes of carrots produced and the amount of fertilizer used. The function <code class='latex inline'>\displaystyle f(x)=-0.53 x^{2}+1.38 x+0.14 </code> models the effect of different amounts of fertilizer, <code class='latex inline'>\displaystyle x </code>, in hundreds of kilograms per hectare <code class='latex inline'>\displaystyle (\mathrm{kg} / \mathrm{ha}) </code>, on the yield of carrots, in tonnes.</p><p>a) Evaluate <code class='latex inline'>\displaystyle f(x) </code> for the given values of <code class='latex inline'>\displaystyle x </code> and complete the table.</p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline Fertilizer, \boldsymbol{x}(\mathbf{k g} / \mathrm{ha}) & 0.00 & 0.25 & 0.50 & 0.75 & 1.00 & 1.25 & 1.50 & 1.75 & 2.00 \\ \hline Yield, \boldsymbol{y}(\boldsymbol{x}) (tonnes) & & & & & & & & & \\ \hline \end{array} </code></p><p>b) According to the table, how much fertilizer should the farmers use to produce the most tonnes of carrots?</p><p>c) Check your answer with a graphing calculator or by evaluating <code class='latex inline'>\displaystyle f(x) </code> for values between <code class='latex inline'>\displaystyle 1.25 </code> and <code class='latex inline'>\displaystyle 1.50 </code>. Why does the answer change? Explain.</p>
<p>The stopping distance for a boat in calm water is modelled by the function <code class='latex inline'>d(v)=0.004v^2+0.2v+6</code>, where <code class='latex inline'>d(v)</code> is in metres and <code class='latex inline'>v</code> is in kilometres per hour.</p><p>What is the initial speed of the boat if it takes <code class='latex inline'>11.6</code> m to stop?</p>
<img src="/qimages/46092" /><p>MODELING WITH MATHEMATICS You design a frame to surround a rectangular photo. The width of the frame is the same on every side, as shown.</p><p>a. Write a polynomial that represents the combined area of the photo and the frame.</p><p>b. Find the combined area of the photo and the frame when the width of the frame is 4 inches.</p>
<p>The actuarial firm where Andrea has her co-op placement has sent a set of dat that follows a quadratic function. The data supplied compared the number of years of driving experience with the number of collisions reported to an insurance company in the last month. Andrew was asked to recover the data lost when the paper jammed in the fax machine. Only three data points can be read. They are <code class='latex inline'>(4, 22), (8, 28)</code>, and <code class='latex inline'>(9, 22)</code>. The values of <code class='latex inline'>f(x)</code> for <code class='latex inline'>x = 6</code> and <code class='latex inline'>x= 7</code> are missing. Andrew decided to subtract the <code class='latex inline'>y-</code>value of 22 from each point so that she would have two zeros: <code class='latex inline'>(5, 0), (8, 6)</code>, and <code class='latex inline'>(9, 0)</code>.</p><p><strong>(a)</strong> Use these three points to find a quadratic function that can be used to model the adjusted data.</p>
<p> A school&#39;s rectangular athletic fields currently have a length of 125 yd and a width of 75 yd. The school plans to expand both the length and the width of the fields by <code class='latex inline'> x </code> yards. What polynomial in standard form represents the area of the expanded athletic field?</p>
<p>A toy rocket is placed on a tower and launched straight up. The table shows its height, <code class='latex inline'>y</code>, in metres above the ground after <code class='latex inline'>x</code> seconds.</p><img src="/qimages/9869" /><p>a) What is the height of the tower?</p><p>b) How long is the rocket in flight?</p><p>c) Do the data in the table represent a quadratic relation? Explain.</p><p>d) Create a scatter plot. Then draw a curve of good fit.</p><p>e) Determine the equation of your curve of good fit.</p><p>f) What is the maximum height of the rocket?</p>
<p>Business The weekly revenue for a company is <code class='latex inline'>\displaystyle r=-3 p^{2}+60 p+1060 </code>, where <code class='latex inline'>\displaystyle p </code> is the price of the company&#39;s product. Use the discriminant to find whether there is a price for which the weekly revenue would be <code class='latex inline'>\displaystyle \$ 1500 </code>.</p>
<p>Sports A diver dives from a <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> springboard. The equation <code class='latex inline'>\displaystyle f(t)=-4.9 t^{2}+4 t+10 </code> models her height above the pool at time <code class='latex inline'>\displaystyle t </code> in seconds. At what time does she enter the water?</p>
<ol> <li>Physics When serving in tennis, a player tosses the tennis ball vertically in the air. The height <code class='latex inline'>\displaystyle h </code> of the ball after <code class='latex inline'>\displaystyle t </code> seconds is given by the quadratic function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+7 t </code> (the height is measured in meters from the point of the toss).</li> </ol> <p>a. How high in the air does the ball go?</p><p>b. Assume that the player hits the ball on its way down when it&#39;s <code class='latex inline'>\displaystyle 0.6 \mathrm{~m} </code> above the point of the toss. For how many seconds is the ball in the air between the toss and the serve?</p>
<p>The parks department is planning a new flower bed outside city hall. It will be rectangular with dimensions 9 m by 6 m. The flower bed will be surrounded by a path of constant width with the same area as the flower bed. Calculate the perimeter of the outside of the path.</p><img src="/qimages/1878" /><p>a) Set up a quadratic equation to model the question.</p><p>b) Use the quadratic formula to solve the problem.</p><p>c) Check your solution using a graphing calculator.</p>
<p>The daily production cost, <code class='latex inline'>C</code>, of a special- edition toy car is given by the function <code class='latex inline'>C(t) = 0.2t^2 - 10t + 650</code>, where <code class='latex inline'>C(t)</code> is in dollars and <code class='latex inline'>t</code> is the number of cars made.</p><p>a) How many cars must be made to minimize the production cost?</p><p>b) Using the number of cars from part (a), determine the cost.</p>
<p>A tenpin bowling lane is 17m longer than it is wide. The area of the lane is <code class='latex inline'>18 m^2</code>. What are the dimensions of the lane?</p>
<p>It seems that the sum of the squares of two consecutive even or odd integers is always even. For example: </p><p><code class='latex inline'>4^2+6^2=16+36</code></p><p><code class='latex inline'>=52</code></p><p><code class='latex inline'>7^2+9^2=49+81</code></p><p><code class='latex inline'>=130</code></p><p>Let <code class='latex inline'>n</code> represent the first integer.</p><p>a) What expression represents the second integer?</p><p>b) What expression represents the sum of the two squares?</p><p>c) Use algebra to show that the result is always even.</p>
<p>In the problem about selling raffle tickets in Getting Started on page 5, the student council wants to determine the price of the tickets. The council surveyed students to find out how many tickets would be bought at different prices. The council found that</p> <ul> <li><p>if they charge <code class='latex inline'>\displaystyle \$ 0.50 </code>, they will be able to sell 200 tickets; and</p></li> <li><p>if they raise the price to <code class='latex inline'>\displaystyle \$ 1.00 </code>, they will sell only 50 tickets.</p></li> </ul> <p>a) The formula for revenue, <code class='latex inline'>\displaystyle R(x) </code>, as a function of ticket price, <code class='latex inline'>\displaystyle x </code>, is <code class='latex inline'>\displaystyle R(x)=-300 x^{2}+350 x . </code> Use a graphing calculator to graph the function.</p><p>b) What ticket price should they charge to generate the maximum amount of revenue?</p><p>c) State the range and domain of the function.</p>
<p>The parabolic shape of new bridge in Calgary can be approximated by the equation <code class='latex inline'> h = - \dfrac{1}{25}x^2 + \dfrac{16}{5}x</code>, where <em>x</em> is the horizontal distance, in metres, from one end and <em>h</em> is the height, in metres, above the water. </p><p>a) Graph the quadratic relation with or without technology. </p><p>b) What is the height of the bridge 15 m horizontally from one end?</p><p>c) How wide is the bridge at its base?</p><p>d) What is the maximum height of the bridge? At what horizontal distance does it reach that height?</p><p>e) Identify the axis of symmetry of the bridge?</p>
<ol> <li>Retail sales A sporting goods store sells 90 ski jackets in a season for <code class='latex inline'>\displaystyle \$ 200 </code> each. Each <code class='latex inline'>\displaystyle \$ 10 </code> decrease in the price would result in five more jackets being sold.</li> </ol> <p>a) Find the number of jackets sold and the selling price to give revenues of <code class='latex inline'>\displaystyle \$ 17600 </code> from sales of ski jackets. b) What is the lowest price that would produce revenues of at least <code class='latex inline'>\displaystyle \$ 15600 </code> ? How many jackets would be sold at this price?</p>
<p>A bus company usually transports 12 000 people per day at a ticket price of <code class='latex inline'>\$1</code>. The company wants to raise the ticket price. For every <code class='latex inline'>\$0.10</code> increase in the ticket price, ,the number of riders per day is expected to decrease by 400. Calculate the ticket price that will maximize revenue.</p>
<p>A model rocket is launched straight up, with an initial velocity of <code class='latex inline'>150 m/s</code>. The height of the rocket can be modelled by <code class='latex inline'>h=-5t^2+150t</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the elapsed time in seconds. What is the maximum height reached by the rocket?</p>
<p>Determine the maximum area of a</p><p>triangle, in square centimetres, if the sum of its base and its height is <code class='latex inline'>\displaystyle 15 \mathrm{~cm} . </code></p>
<p>A rectangular swimming pool measuring 10m by 4m is surrounded by a deck of uniform width. The total area of the pool and deck is 135 <code class='latex inline'>m^2</code>. What is the width of the deck?</p><img src="/qimages/9422" />
<p>A dance club has a <code class='latex inline'>\$5</code> cover charge and averages 300 customers on Friday nights. Over the past several months, the club has changed the cover price several times to see how this affects the number of customers. For every increase of <code class='latex inline'>\$0.50</code> in the cover charge, the number of customers decreases by 30. Use an algebraic model to determine the cover charge that maximizes revenue.</p>
<p>The path of a soccer ball after it is kicked from a height of 0.5 m above the ground is given by the equation <code class='latex inline'>h=-0.1d^2+d+0.5</code>, where <code class='latex inline'>h</code> is the height. in metres, above the ground and <code class='latex inline'>d</code> is the horizontal distance, in metres. </p> <ul> <li> How far has the soccer ball travelled horizontally. to the nearest tenth of a metre, when it lands on the ground?</li> </ul>
<p>A model airplane is shot into the air. Its path is approximated by the function <code class='latex inline'>h(t) = - 5t^2 + 25t</code>, where <code class='latex inline'>h(t)</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds. When will the airplane hit the ground?</p>
<p>For the relation, explain what each coordinate of the vertex represents and what the zeros represent.</p> <ul> <li>a relation that models the profit earned, <code class='latex inline'>P</code>, on an item at a given selling price, <code class='latex inline'>s</code></li> </ul>
<p>A sky diver jumped from an airplane. He used his watch to time the length of his jump. His height above the ground can be modelled by <code class='latex inline'>h=-5(t-4)^2+2500</code>, where <code class='latex inline'>h</code> is his height above the ground in metres and t is the time in seconds from the time he started the timer.</p><p><strong>a)</strong> How long did the sky diver delay his jump?</p><p><strong>b)</strong> From what height did he jump?</p>
<ol> <li>Geometry A regular polygon with <code class='latex inline'>\displaystyle n </code> sides has <code class='latex inline'>\displaystyle \frac{n(n-3)}{2} </code> diagonals. Find the number of sides of a regular polygon that has 44 diagonals.</li> </ol>
<ol> <li>Measurement The length of a rectangle is <code class='latex inline'>\displaystyle 5 \mathrm{~cm} </code> greater than twice the width. The area is <code class='latex inline'>\displaystyle 33 \mathrm{~cm}^{2} </code>. Find the dimensions.</li> </ol>
<p>A rectangular carpet and a square carpet have equal areas. The square carpet has a side length of 4 m. The length of the rectangular carpet is 2 m less than three times its width. Find the dimensions of the rectangular carpet. </p>
<p>The sum of the first <em>n</em> even natural numbers is a quadratic relation. Determine that relation and verify it for the first six even natural numbers. </p>
<p>Three times the square of an integer is 432. Find the integer. </p>
<p>A basketball shot is taken from a horizontal distance of <code class='latex inline'>5</code> m from the hoop. The height of the ball can be modelled by the relation <code class='latex inline'>h = -7.3t^2 + 8.25t + 2.1</code>, where <code class='latex inline'>h</code> is the height, in metres, and <code class='latex inline'>t</code> is the time, in seconds, since the ball was released.</p><p>a) From what height was the ball released?</p><p>b) What was the maximum height reached by the ball?</p><p>c) If the ball reached the hoop in <code class='latex inline'>1</code>s, what was the height of the hoop?</p>
<p>A firecracker is fired from the ground. The height of the firecracker at a given time is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+50 t </code>, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. When will the firecracker reach a height of <code class='latex inline'>\displaystyle 45 \mathrm{~m} </code> ?</p>
<p>The specifications for a cardboard box state that the width is 5 cm less than the length, and the height is 1 cm more than double the length. Write an equation for the volume of the box and find possible dimensions for a volume of 550 <code class='latex inline'>cm^3</code>.</p><p><strong>You may use a graphing device for this question</strong></p>
<p>Write an expression for the area of each shaded region as a polynomial and then in factored form. </p><p>0</p>
<ol> <li>Measurement The height of a triangle is 2 units more than the base. The area of the triangle is 10 square units. Find the base, to the nearest hundredth.</li> </ol>
<p>A police officer has <code class='latex inline'>\displaystyle 400 \mathrm{~m} </code> of yellow tape to seal off the area of a crime scene. What is the maximum area that can be enclosed?</p>
<p>Conor has a summer lawn-mowing business. Based on experience, Conor knows that <code class='latex inline'>P=-5x^2+200x-1500</code> models his profit, <code class='latex inline'>P</code>, in dollars, where <code class='latex inline'>x</code> is the amount, in dollars, charged per lawn.</p> <ul> <li>How much does he need to charge if he wants to have a profit of $500?</li> </ul>
<ol> <li>Measurement A triangle has a height of <code class='latex inline'>\displaystyle 6 \mathrm{~cm} </code> and a base of <code class='latex inline'>\displaystyle 8 \mathrm{~cm} </code>. If the height and the base are both decreased by the same amount, the area of the new triangle is <code class='latex inline'>\displaystyle 20 \mathrm{~cm}^{2} </code>. What are the base and height of the new triangle, to the nearest tenth of a centimetre?</li> </ol>
<p>The relation <code class='latex inline'>\displaystyle d = 0.0052s^2 + 0.13s </code> models the stopping distance, <code class='latex inline'>d</code>, in metres, of a car travelling at a speed of <code class='latex inline'>s</code>, in kilometres per hour, when the driver brakes hard. At what speed was a car travelling if its stopping distance is 20m? Round to the nearest tenth?</p>
<p>The actuarial firm where Andrea has her co-op placement has sent a set of dat that follows a quadratic function. The data supplied compared the number of years of driving experience with the number of collisions reported to an insurance company in the last month. Andrew was asked to recover the data lost when the paper jammed in the fax machine. Only three data points can be read. They are <code class='latex inline'>(4, 22), (8, 28)</code>, and <code class='latex inline'>(9, 22)</code>. The values of <code class='latex inline'>f(x)</code> for <code class='latex inline'>x = 6</code> and <code class='latex inline'>x= 7</code> are missing. Andrew decided to subtract the <code class='latex inline'>y-</code>value of 22 from each point so that she would have two zeros: <code class='latex inline'>(5, 0), (8, 6)</code>, and <code class='latex inline'>(9, 0)</code>.</p><p><strong>(b)</strong> Add a <code class='latex inline'>y-</code>value of 22 to this function for a quadratic function that models the original data.</p><p><strong>(c)</strong> Use this function to find the missing values for <code class='latex inline'>x = 6</code> and <code class='latex inline'>x = 7</code>.</p>
<p>The height above the ground of a bungee jumper is modelled by the aquadratic function <code class='latex inline'>h(t) = -5(t - 0.3)^2+110</code> where height, <code class='latex inline'>h(t)</code>, is in metres and time, t, is in seconds.</p><p>a) When does the bungee jumper reach maximum height? Why is it a maximum?</p><p>b) What is the maximum height reached by the jumper?</p><p>c) Determine the height of the platform from which the bungee jumper jumps.</p>
<p>A rectangle has an area of <code class='latex inline'>\displaystyle 6 x^{2}-8 . </code> a) Determine the dimensions of the rectangle. b) Is there more than one possibility? Explain.</p>
<p>The height of an arrow shot on Neptune can be modelled by the quadratic function <code class='latex inline'>h(t)=2.3+50t-5.57t^2</code>, where time, <code class='latex inline'>t</code>, is in seconds and height, <code class='latex inline'>h(t)</code> , is in metres. Use the quadratic formula to determine when the arrow will hit the surface.</p>
<p>A rapid—transit company has 5000 passengers daily, each currently paying a $2.25 fare. For each $0.50 increase, the company estimates that it will lose 150 passengers daily. If the company must be paid at least $15 275 each day to stay in business, what minimum fare must they charge to produce this amount of revenue?</p>
<p>The platforms on the ends of the half-pipe are at the same height. </p><img src="/qimages/1118" /><p>a) How wide is the half-pipe?</p><p><code class='latex inline'>\to</code> b) How far would a skater have travelled horizontally after a drop of 2 m? Round to the nearest hundredth of a metre.</p>
<img src="/qimages/46150" /><p>MODELING WITH MATHEMATICS In deer, the gene <code class='latex inline'>\displaystyle N </code> is for normal coloring and the gene <code class='latex inline'>\displaystyle a </code> is for no coloring, or albino. Any gene combination with an <code class='latex inline'>\displaystyle N </code> results in normal coloring. The Punnett square shows the possible gene combinations of an offspring and the resulting colors from parents that both have the gene combination Na.( See Example 4.)</p><p>a. What percent of</p><p>the possible gene</p><p>combinations</p><p>result in albino</p><p>coloring?</p><p>b. Show how you</p><p>could use a</p><p>polynomial</p><p>to model the</p><p>possible gene</p><p>combinations</p><p>of the offspring.</p>
<p>The safe stopping distance for a boat travelling at a constant speed in calm water is given by <code class='latex inline'>\displaystyle d(v)=0.002\left(2 v^{2}+10 v+3000\right) </code>, where <code class='latex inline'>\displaystyle d(v) </code> is the distance in metres and <code class='latex inline'>\displaystyle v </code> is the speed in kilometres per hour. What is the initial speed of the boat if it takes <code class='latex inline'>\displaystyle 30 \mathrm{~m} </code> to stop?</p>
<p>Sara has designed a fishpond in the shape of a right triangle with two sides of length <code class='latex inline'>a</code> and <code class='latex inline'>b</code> and hypotenuse of length <code class='latex inline'>c</code>.</p><p>Write an expression in factored form for <code class='latex inline'>a^2</code>.</p>
<p>The profit of a shoe company is modelled by the quadratic function <code class='latex inline'>P(x)=-5(x-4)^2+45</code>, where <code class='latex inline'>x</code> is the number of pairs of shoes produced, in thousands, and <code class='latex inline'>P(x)</code> is the profit, in thousands of dollars. How many thousands of pairs of shoes will the company need to sell to earn a profit?</p>
<p>A square flower garden is surrounded by a brick walkway that is 1.5 m wide. The area of the walkway is equal to the area of the garden. Determine the dimensions of the flower garden, to the nearest tenth of a metre.</p>
<ol> <li>Currency In Britain, people refer to their paper money as notes. A five-pound note is worth 5 pounds sterling. The length of a five-pound note is <code class='latex inline'>\displaystyle 5 \mathrm{~mm} </code> less than twice the width. The area is <code class='latex inline'>\displaystyle 9450 \mathrm{~mm}^{2} </code>. a) Find the dimensions of a five-pound note. b) Communication How do the dimensions compare with those of a Canadian <code class='latex inline'>\displaystyle \$ 20 </code> bill?</li> </ol>
<p>The function <code class='latex inline'>\displaystyle h(t)=2.3+50 t-1.86 t^{2} </code> models the height of an arrow shot from a bow on Mars, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. How long does the arrow stay in flight?</p>
<p>The arch of a bridge is modelled by the function <code class='latex inline'>h(d) = 2-0.043d^2+2.365d</code>, where h is the height, in metres, and dis the horizontal distance, in metres, from the origin of the arch.</p><p>a) Determine the maximum height of the arch, to the nearest hundredth of a metre.</p><p>b) What is the width of the arch at its base?</p>
<ol> <li>Diving Annie Pelletier won a bronze medal for Canada in women&#39;s springboard diving at the Summer Olympics in Atlanta. She dove from a springboard with dimensions that can be represented by the binomials <code class='latex inline'>\displaystyle 7 x-2 </code> and <code class='latex inline'>\displaystyle x-10 . </code> a) Multiply the binomials.</li> </ol> <p>b) If <code class='latex inline'>\displaystyle x </code> represents <code class='latex inline'>\displaystyle 70 \mathrm{~cm} </code>, what was the area of the board, in square centimetres? in square metres?</p>
<p>What dimensions can a rectangle with an area of <code class='latex inline'>\displaystyle 12 x^{2}-3 x-15 </code> have?</p>
<p>At the traffic safety bureau, Paul is conducting a study on the stoplights at a particular intersection. He determines that when there are <code class='latex inline'>18</code> green lights per hour, then, on average, <code class='latex inline'>12</code> cars can safely travel through the intersection on each green light. He also finds that if the number of green lights per hour increases by one, then one fewer car can travel through the intersection per light.</p><p>How many green lights should there be per hour to maximize the number of cars through the intersection?</p>
<p>The length of the hypotenuse of a right triangle is 1 cm more than triple that of the shorter leg. The length of the longer leg is 1 cm less than triple that of the shorter leg. Find the lengths of the three sides of the triangle.</p>
<p>The St. Louis Gateway Arch in St. Louis, Missouri. was built in 1005 and was designed as a catenary. which is a curve that approximates a parabola. The arch is 102 m wide and 102 m tall.</p><p>a) Sketch a graph of the arch that is symmetrical about the y-axis.</p><p>b) Label the x-intercepts and the vertex.</p><p>c) Determine an equation to model the arch.</p>
<p>A football is kicked into the air. Its height above the ground is approximated by the relation <code class='latex inline'>h=20t-5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds since the football was kicked.</p> <ul> <li>What are the coordinates of the vertex?</li> </ul>
<p>A farmer enclosed a rectangular field</p><p>with <code class='latex inline'>\displaystyle 400 \mathrm{~m} </code> of fencing. The area of the field is <code class='latex inline'>\displaystyle 9000 \mathrm{~m}^{2} </code>. Determine the dimensions of the field.</p>
<p>The hypotenuse of a right triangle measures 20 cm. The sum of the lengths of the legs is 28 cm. Find the length of each leg of the triangle.</p>
<p>A circle with diameter 8 cm is removed from a larger circle with radius <code class='latex inline'>r</code> cm.</p><img src="/qimages/5224" /><p>a) Express the area of the shaded region as a function of <code class='latex inline'>r</code>.</p><p>b) Express the area of the shaded region in factored form.</p><p>c) State the domain and range of the area function.</p>
<p>The population of a city is modelled by the function <code class='latex inline'>\displaystyle P(t)=0.5 t^{2}+10 t+200 </code>, where <code class='latex inline'>\displaystyle P(t) </code> is the population in thousands and <code class='latex inline'>\displaystyle t </code> is time in years. Note: <code class='latex inline'>\displaystyle t=0 </code> corresponds to the year 2000 . According to the model, when will the population reach 312000 ?</p>
<p>A company that manufactures MP3 players uses the relation <code class='latex inline'>P=120x-60x^2</code> to model its profit. The variable <code class='latex inline'>x</code> represents the number of thousands of MP3 players sold. The variable <code class='latex inline'>P</code> represents the profit in thousands of dollars.</p> <ul> <li>The company &quot;breaks even&quot; when the profit is zero. Are there any break-even points for this company? If so, how many MP3 player&#39;s are sold at the break-even points?</li> </ul>
<p>The sum of the squares of two consecutive odd integers is 290. Find the integers.</p>
<ul> <li>13. Consider the domed sports arena</li> </ul> <p>described in question 12. Suppose that</p><p>instead of having the vertex on the <code class='latex inline'>\displaystyle y </code>-axis, the parabola was positioned with one end of the arch at the origin of the grid.</p><p>a) Determine the equation, in factored</p><p>form and in standard form, of</p><p>the quadratic function for this</p><p>orientation.</p><p>b) Find the maximum height of the</p><p>arch and compare the result to the</p><p>height calculated in question <code class='latex inline'>\displaystyle 12 . </code></p>
<ol> <li>Measurement The surface area of a cube is <code class='latex inline'>\displaystyle 384 \mathrm{~cm}^{2} </code>. Find the length of one edge.</li> </ol>
<p>The student council is selling cases of gift cards as a fundraiser. The revenue, <code class='latex inline'>\displaystyle R(x) </code>, in dollars, can be modelled by the function <code class='latex inline'>\displaystyle R(x)=-25 x^{2}+100 x+1500 </code>, where <code class='latex inline'>\displaystyle x </code> is the number of cases of gift cards sold. How many cases must the students sell to maximize their revenue?</p>
<p>The path of a toy rocket is modelled by the equation <code class='latex inline'>y=-x^2+6x+2</code>, where <code class='latex inline'>x</code> is the horizontal distance, in metres, travelled and <code class='latex inline'>y</code> is the height, in metres, of the toy rocket above the ground. What is the maximum height of the toy rocket? At what horizontal distance does the maximum height occur?</p>
<ol> <li>Basketball court The width of a basketball court is <code class='latex inline'>\displaystyle 1 \mathrm{~m} </code> more than half the length. If the area of the court is <code class='latex inline'>\displaystyle 364 \mathrm{~m}^{2} </code>, find the length and the width.</li> </ol>
<p>Ryan owns small music store. He currently charges <code class='latex inline'>\$10</code> for each CD. At this price, he sells about <code class='latex inline'>80</code> CDs a week. Experience has taught him that a $1 increase in the price of a CD means a drop of about five CDs per week in sales. At what price should Ryan sell his CDs to maximize his revenue?</p>
<p> The sum of two numbers is 24 and the sum of their squares is 306. What are the numbers?</p>
<p>A square lawn is surrounded by a concrete walkway that is <code class='latex inline'>\displaystyle 2.0 \mathrm{~m} </code> wide, as shown at the left. If the area of the walkway equals the area of the lawn, what are the dimensions of the lawn? Express the dimensions to the nearest tenth of a metre.</p><img src="/qimages/15026" />
<p> The stopping distance <code class='latex inline'>d</code>, in metres, of a car travelling at a velocity of <code class='latex inline'>v</code> km/h is given by the formula <code class='latex inline'>d = 0.007v^2 + 0.015v</code>.</p><p>How fast, to the nearest whole number, is a car travelling if it takes 30 m to stop?</p>
<ol> <li>Hedge maze The world&#39;s largest hedge maze is in the grounds of an English country house known as Longleat. The rectangular maze has <code class='latex inline'>\displaystyle 2.7 \mathrm{~km} </code> of paths flanked by 16180 yew trees. The length of the rectangle is <code class='latex inline'>\displaystyle 60 \mathrm{~m} </code> more than the width. The area of the rectangle is <code class='latex inline'>\displaystyle 6496 \mathrm{~m}^{2} </code>. What are the dimensions of the rectangle?</li> </ol>
<p>Jasmine and Raj have <code class='latex inline'>\displaystyle 24 \mathrm{~m} </code> of fencing to enclose a rectangular garden. What are the dimensions of the largest rectangular garden they can enclose with that length of fencing?</p>
<ol> <li>Fenced field The area of a rectangular field is <code class='latex inline'>\displaystyle 2275 \mathrm{~m}^{2} </code>. The field is enclosed by <code class='latex inline'>\displaystyle 200 \mathrm{~m} </code> of fencing. What are the dimensions of the field?</li> </ol>
<p>Cam has 46 m of fencing to enclose a meditation space on the grounds of his local hospital. He has decided that the meditation space should be rectangular, with fencing on only three sides. What dimensions will give the patients the maximum amount of meditation space?</p>
<ol> <li>Measurement The hypotenuse of a right triangle has a length of <code class='latex inline'>\displaystyle 13 \mathrm{~cm} </code>. The sum of the lengths of the other two sides is <code class='latex inline'>\displaystyle 17 \mathrm{~cm} </code>. Find the unknown side lengths.</li> </ol>
<ol> <li>Touch football A touch football quarterback passed the ball to a receiver <code class='latex inline'>\displaystyle 40 \mathrm{~m} </code> downfield. The path of the ball can be described by the function</li> </ol> <p><code class='latex inline'>\displaystyle h=-0.01(d-20)^{2}+6 </code></p><p>where <code class='latex inline'>\displaystyle h </code> is the height of the ball, in metres, and <code class='latex inline'>\displaystyle d </code> is the horizontal distance of the ball from the quarterback, in metres. a) What was the maximum height of the ball? b) What was the horizontal distance of the ball from the quarterback at its maximum height?</p><p>c) What was the height of the ball when it was thrown? when it was caught? d) If a defensive back was <code class='latex inline'>\displaystyle 2 \mathrm{~m} </code> in front of the receiver, how far was the defensive back from the quarterback?</p><p>e) How high would the defensive back have needed to reach to knock down the pass?</p>
<p>Determine when the diver is <code class='latex inline'>5</code> m above the water.</p><p>A cliff diver dives from about <code class='latex inline'>17</code> m above the water. The divers height above the water. <code class='latex inline'>h(t)</code>,in metres, after <code class='latex inline'>t</code> seconds is modelled by <code class='latex inline'>h(t)=-4.9t^2+1.5t+17</code>. Explain how to determine when the diver is <code class='latex inline'>5</code> m above the water.</p>
<img src="/qimages/90117" />
<p>A trained stunt diver is diving off a platform that is <code class='latex inline'>\displaystyle 15 \mathrm{~m} </code> high into a pool of water that is <code class='latex inline'>\displaystyle 45 \mathrm{~cm} </code> deep. The height, <code class='latex inline'>\displaystyle h </code>, in metres, of the stunt diver above the water is modelled by <code class='latex inline'>\displaystyle h=-4.9 t^{2}+1.2 t+15 </code>, where <code class='latex inline'>\displaystyle t </code> is the time in seconds after starting the dive. a) How long is the stunt diver above <code class='latex inline'>\displaystyle 15 \mathrm{~m} </code> ?</p>
<p>The perimeter of a rectangle is 8 m and its area is <code class='latex inline'>2 m^2</code>. Find the length and width of the rectangle to the nearest tenth of a metre.</p>
<p>When the square of an integer is added to ten times the integer, the sum is zero, What is the integer?</p>
<p>A gardener wants to fence three sides of the yard in front of her house. She bought <code class='latex inline'>60</code> m of fence and wants an area of about <code class='latex inline'>400</code> m<code class='latex inline'>^2</code>. The quadratic equation <code class='latex inline'>f(x)=60x-2x^2</code>, where <code class='latex inline'>x</code> is the width of the yard in metres and <code class='latex inline'>f(x)</code> is the area in square metres, gives the area that can be enclosed. Determine the dimensions that will give the desired area.</p>
<ol> <li>Measurement The height of a triangle is <code class='latex inline'>\displaystyle 2 \mathrm{~m} </code> more than the base. The area is <code class='latex inline'>\displaystyle 17.5 \mathrm{~m}^{2} </code>. Find the length of the base.</li> </ol>
<p>Target-shooting disks are launched into the air from a machine <code class='latex inline'>\displaystyle 12 \mathrm{~m} </code> above the ground. The height, <code class='latex inline'>\displaystyle h(t) </code>, in metres, of the disk after launch is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+30 t+12 </code>, where <code class='latex inline'>\displaystyle t </code> is time in seconds.</p><p>a) When will the disk reach the ground?</p><p>b) What is the maximum height the disk reaches?</p>
<p>A ball is thrown upward at an initial velocity of 30 m/s, from a height of 2 m. The height, <em>h</em>, in metres, of the ball above the ground after <em>t</em> seconds, can be found using the relation <code class='latex inline'> h = -4.9t^2 + 30t +2</code>. </p><p>a) Graph this relation using a graphing calculator. </p><p>b) Describe the relationship between time and height. </p><p>c) Repeat parts a) and b) for a ball thrown upward on Mars, with a height defined by the relation <code class='latex inline'>h = -1.8372t^2 = 30t +2</code>. </p><p>d) Repeat parts a) and b) for a ball thrown upward from Neptune, with a height defined by the relation <code class='latex inline'>h = -7.007t^2 + 30t +2</code>. </p><p>e) Compare the results from the three locations. </p>
<ol> <li>Baseball The following function gives the height, <code class='latex inline'>\displaystyle h </code> metres, of a batted baseball as a function of the time, <code class='latex inline'>\displaystyle t </code> seconds, since the ball was hit. <code class='latex inline'>\displaystyle h=-6(t-2.5)^{2}+38.5 </code></li> </ol> <p>a) What was the maximum height of the ball? b) What was the height of the ball when it was hit? c) How many seconds after it was hit did the ball hit the ground, to the nearest second?</p><p>d) Find the height of the ball <code class='latex inline'>\displaystyle 1 \mathrm{~s} </code> after it was hit.</p>
<p>Three pieces of a rod measure 20 cm, 41 cm, and 44 cm. If the same amount is cut off from each piece, the remaining lengths can be formed into a right triangle. Determine the length that should be cut off each piece.</p>
<p>Snowy’s Snowboard Co. manufactures snowboards. The company uses the function <code class='latex inline'>P(x) = 324x - 54x^2</code> to model its profit, where<code class='latex inline'>P(x)</code> is the profit in thousands of dollars and <code class='latex inline'>x</code> is the number of snowboards sold, in thousands.</p><p>a) How many snowboards must be sold for the company to break even?</p><p>b) How many snowboards must be sold for the company to be profitable?</p>
<p>The cost per hour of funning an assembly line in a manufacturing plant is a function of the number of items produced per hour. The cost function is <code class='latex inline'>C(x) = 0.3x^2 -1.2x +2</code>, where <code class='latex inline'>C(x)</code> is the cost per hour in thousands of dollars, and x is the number of items produced per hour, in thousands. Determine the most economical production level.</p>
<ol> <li>Numbers Two numbers differ by 6 . If the numbers are squared and then added, the result is 146 . What are the numbers?</li> </ol>
<p>The shape of the Humber River pedestrian bridge in Toronto can be modelled by the equation <code class='latex inline'>y=-0.0044x^2+21.3</code>. All measurements are in metres. Determine the length of the bridge and the maximum height above the ground, to the nearest tenth of a metre.</p>
<p>A golf ball is hit and its height is given by the equation <code class='latex inline'>\displaystyle h=29.4 t-4.9 t^{2} </code>, where <code class='latex inline'>\displaystyle t </code> is the time elapsed, in seconds, and <code class='latex inline'>\displaystyle h </code> is the height, in metres.</p><p>a) Write an equation in function notation to represent the height of the ball as a function of time.</p><p>b) State the degree of this function and whether the function is linear or quadratic.</p><p>c) Use difference tables to confirm your answer in part (b).</p><p>d) Graph the function where <code class='latex inline'>\displaystyle \{t \in \mathbf{R} \mid t \geq 0\} </code>.</p><p>e) At what time(s) is the ball at its greatest height? Express the height of the ball at this time in function notation.</p><p>f) At what time(s) is the ball on the ground? Express the height of the ball at this time in function notation.</p>
<p>The hypotenuse of a right triangle has length 17 cm. The sum of the lengths of the legs is 23 cm. What are their lengths?</p>
<p>A computer software company models the profit on its latest video game with the function <code class='latex inline'>\displaystyle P(x)=-2 x^{2}+32 x-110 </code>, where <code class='latex inline'>\displaystyle x </code> is the number of games the company produces, in thousands, and <code class='latex inline'>\displaystyle P(x) </code> is the profit, in thousands of dollars. How many games must the company sell to make a profit of <code class='latex inline'>\displaystyle \$ 16000 </code> ? a) Write a solution to the problem. Indicate why you chose the strategy you did.</p><p>b) Discuss your solution and reasoning with a partner. Be ready to share your ideas with the class.</p>
<p>A soccer ball is kicked from the ground. After traveling a horizontal distance of <code class='latex inline'>35</code> m, it just passes over a <code class='latex inline'>1.5</code> m tall fence before hitting the ground <code class='latex inline'>37</code> m from where it was kicked.</p><img src="/qimages/574" /><p> Considering the ground to be the <code class='latex inline'>x</code>-axis and the vertex to be on the <code class='latex inline'>y</code>-axis, determine the equation of a quadratic function that can be used to model the parabolic path of the ball.</p>
<p>The cost, in dollars, of operating a machine per day is given by the formula <code class='latex inline'>\displaystyle C = 3t^2-96t+1014 </code>, where t is the time the machine operates, in hours. What is the minimum cost of running the machine? For how many hours must the machine run to reach this minimum cost?</p>
<p> A football is kicked off the ground. After travelling a horizontal distance</p><p>of <code class='latex inline'>\displaystyle 24 \mathrm{~m} </code>, it just passes over a tree that is <code class='latex inline'>\displaystyle 3 \mathrm{~m} </code> tall before hitting the ground <code class='latex inline'>\displaystyle 28 \mathrm{~m} </code> from where it was kicked.</p><p>a) Consider the ground to be the <code class='latex inline'>\displaystyle x </code>-axis and assume the vertex lies on the</p><p><code class='latex inline'>\displaystyle y </code>-axis. Determine the equation of the quadratic function that models the</p><p>path of the ball.</p><p>b) Determine the maximum height of the football.</p><p>c) How far has the football travelled horizontally when it reaches its maximum height?</p><p>d) Suppose the football is kicked from a starting point at the origin. Develop a new equation of the quadratic</p><p>function that represents the football.</p><p>e) Describe the similarities and differences between the functions</p><p>found in parts a and d).</p><p>f) Use Technology Use a graphing calculator and compare the solutions.</p><img src="/qimages/157371" />
<p>The height of an arrow shot by an archer is given by the function <code class='latex inline'>\displaystyle b(t)=-5 t^{2}+18 t-0.25 </code>, where <code class='latex inline'>\displaystyle h(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. The centre of the target is in the path of the arrow and is <code class='latex inline'>\displaystyle 1 \mathrm{~m} </code> above the ground. When will the arrow hit the centre of the target?</p>
<p>The profit, <code class='latex inline'>P(x)</code> , of a video company, in thousands of dollars, is given a by <code class='latex inline'>P(x)=-5x^2+550x-5000</code>, where <code class='latex inline'>x</code> is the amount spent on advertising, in thousands of dollars. Can the company make a profit of $<code class='latex inline'>50\ 000</code>? Explain.</p>
<p>It costs a bus company <code class='latex inline'>\$225</code> to run a minibus on a ski trip, plus <code class='latex inline'>\$30</code> per passenger. The bus has a seating for <code class='latex inline'>22</code> passengers and the company charges <code class='latex inline'>\$60</code> per far if the bus is full. For each empty seat, the company has to increase the ticket price by <code class='latex inline'>\$5</code>. How many empty seats should the bus run with to maximize profit from this trip?</p><p>A. 8</p><p>B. 6</p><p>C. 10</p><p>D. 2</p>
<p>Air Pollution The function <code class='latex inline'>\displaystyle y=0.4409 x^{2}-5.1724 x+99.0321 </code> models the emissions of carbon monoxide in the United States since 1987, where <code class='latex inline'>\displaystyle y </code> represents the amount of carbon monoxide released in a year in millions of tons, and <code class='latex inline'>\displaystyle x=0 </code> represents the year <code class='latex inline'>\displaystyle 1987 . </code></p><p>a. How can you use a graph to estimate the year in which more than 100 million tons of carbon monoxide were released into the air?</p><p>b. How can you use the Quadratic Formula to estimate the year in which more than 100 million tons of carbon monoxide were released into the air?</p><p>c. Which method do you prefer? Explain why.</p>
<p>The president of a company that manufactures toy cars thinks that the function <code class='latex inline'>\displaystyle P(c)=-2 c^{2}+14 c-20 </code> represents the company&#39;s profit, where <code class='latex inline'>\displaystyle c </code> is the number of cars produced, in thousands, and <code class='latex inline'>\displaystyle P(c) </code> is the company&#39;s profit, in hundreds of thousands of dollars. Determine the maximum profit the company can earn.</p>
<p>The population of a town is modelled by the function <code class='latex inline'>P(t)=6t^2+110t+4000</code>, where <code class='latex inline'>P(t)</code> is the population and <code class='latex inline'>t</code> is the time in years since <code class='latex inline'>2000</code>.</p><p>What will the population be in <code class='latex inline'>2020</code>?</p>
<p>You can find the distance in feet that an object falls in <code class='latex inline'>\displaystyle t </code> seconds using the expression <code class='latex inline'>\displaystyle 16 t^{2} </code>. If you drop a ball from a tall building, how far does the ball fall in 3 s?</p><p><code class='latex inline'>\displaystyle \begin{array}{llll}\text { F) } 16 \mathrm{ft} & \text { G } 48 \mathrm{ft} & \text { H } 96 \mathrm{ft} & \text { D } 144 \mathrm{ft}\end{array} </code></p>
<p>A video tracking device recorded the height, f1, in metres, of a baseball after it was hit. The data collected can be modelled by the relation <code class='latex inline'>h=-5(t-2)^2+21</code>, where <code class='latex inline'>t</code> is the time in seconds after the ball was hit.</p> <ul> <li>Approximately when did the baseball hit the ground?</li> </ul>
<p>The triangle with sides given by <code class='latex inline'>x^2 + 1, x^2 -1</code> and <code class='latex inline'>2x</code> will always be right triangle for <code class='latex inline'>x > 1</code>.</p><p>Use the Pythagorean theorem to verify that his statement is true for <code class='latex inline'>x</code>-values of <code class='latex inline'>2</code>, <code class='latex inline'>3</code>, and <code class='latex inline'>4</code>.</p>
<p>The minimum stopping distance, after a delay of 1 s, for a particular car is modelled by the formula <code class='latex inline'>d = 0.006(s + 1)^2</code>, , where <code class='latex inline'>d</code> represents the stopping distance, in metres, and s represents the initial speed, in kilometres per hour.</p><p>a) Expand and simplify the formula.</p><p>b) Compare the results in both versions of the formula for an initial speed of 60 km/h.</p>
<ol> <li>Geometry The area of a square is 3 square units greater than the area of the square shown.</li> </ol> <p>a) Write an equation that relates the area, <code class='latex inline'>\displaystyle A </code>, of the larger square to the value of <code class='latex inline'>\displaystyle x </code>. b) Sketch a graph of <code class='latex inline'>\displaystyle A </code> versus <code class='latex inline'>\displaystyle x </code> for the larger square. c) What value of <code class='latex inline'>\displaystyle x </code> results in the minimum area for the larger square? d) What is the area of the smaller square when the larger square has its minimum area?</p>
<p>A football is kicked into the air. Its height above the ground is approximated by the relation <code class='latex inline'>h=20t-5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds since the football was kicked.</p><p><strong>a)</strong> What are the zeros of the relation? When does the football hit the ground?</p><p><strong>b)</strong> What are the coordinates of the vertex?</p><p><strong>c)</strong> Use the information you found for parts a) and b) to graph the relation.</p>
<p> The height of lava ejected from the Stromboli volcano can be modelled by the relation <code class='latex inline'>h = -5t(t - 11)</code>. where <code class='latex inline'>h</code> is the height, in metres. of the lava above the crater and <code class='latex inline'>t</code> is the time,. in seconds, since it was ejected.</p><p>a) Graph the relation.</p><p>b) Find the maximum height reached by the lava, to the nearest metre.</p><p>c) How long does the lava take to reach the maximum height?</p><p>d) Is the length of time that the lava is in the air twice the answer from part (c)? Explain why or why not.</p>
<p>A quadratic equation has zeros <code class='latex inline'>-2</code> and <code class='latex inline'>6</code> and bases through the point <code class='latex inline'>(3, 15)</code>.</p><p><strong>(a)</strong> Find the equation of the quartic function in factored form.</p><p><strong>(b)</strong> Write the function in standard form.</p><p><strong>(c)</strong> Complete the square to convert the standard form to vote form, and state the vertex.</p><p><strong>(d)</strong> Use partial factoring to verify your answer s to part (c).</p><p><strong>(e)</strong> Find a second quadratic function with the same zeros as in part (a), but passing through the point <code class='latex inline'>(3, -30)</code>. Express the function in standard form.</p><p><strong>(f)</strong> Graph both functions. Explain how the graphs can be used to verify that the equations in part (a) and (3) are correct.</p>
<ol> <li>Geometry a) Write an equation that relates the area of a circle, <code class='latex inline'>\displaystyle A </code>, to its radius, <code class='latex inline'>\displaystyle r </code>.</li> </ol> <p>b) Graph <code class='latex inline'>\displaystyle A </code> versus <code class='latex inline'>\displaystyle r </code>. c) Communication Does the graph have an axis of symmetry? Explain. d) State the domain and range of the function.</p>
<p>A circle of radius 3 cm is removed form a circle of raids <code class='latex inline'>r</code>.</p><img src="/qimages/795" /><p><strong>(a)</strong> Express the area of the shaded region as a function of <code class='latex inline'>r</code>.</p><p><strong>(b)</strong> State the domain and range of the area function.</p>
<p> The diagrams show the first three rectangles in a pattern.</p><p>a) State the area of the 4th rectangle.</p><p>b) Write a product of two binomials to represent the area of the <code class='latex inline'>\displaystyle n </code> th rectangle in terms of <code class='latex inline'>\displaystyle n . </code> c) Multiply the binomials from part b).</p><p>d) State the area of the 28 th rectangle, in square centimetres.</p><img src="/qimages/155605" />
<ol> <li>Television screens The size of a television screen or a computer monitor is usually stated as the length of the diagonal. A screen has a <code class='latex inline'>\displaystyle 38-\mathrm{cm} </code> diagonal. The width of the screen is <code class='latex inline'>\displaystyle 6 \mathrm{~cm} </code> more than the height. Find the dimensions of the screen, to the nearest tenth of a centimetre.</li> </ol>
<p>A football is kicked into the air. Its height above the ground is approximated by the relation <code class='latex inline'>h=20t-5t^2</code>, where <code class='latex inline'>h</code> is the height in metres and <code class='latex inline'>t</code> is the time in seconds since the football was kicked.</p> <ul> <li>What are the zeros of the relation? When does the football hit the ground?</li> </ul>
<p>The Rudy Snow Company makes custom snowboards. The company’s profit can be modelled with the relation <code class='latex inline'> \displaystyle y = -6x^2 + 42x - 60 </code>, where x is the number of snowboards sold (in thousands) and <code class='latex inline'>y</code> is the profit (in hundreds of thousands of dollars).</p><p>a. How many snowboards does the company need to sell to break even?</p><p>b. How many snowboards does the company need to sell to maximize their profit?</p>
<p>When a car is traveling at a given speed, there is a minimum turn radius it can safely make. A particular car’s minimum radius can be calculated by <code class='latex inline'>r = 0.6s^2</code>, where <code class='latex inline'>s</code> is the speed, in kilometres per hour, and <code class='latex inline'>r</code> is the turning radius, in metres. If the car uses tires with better grip, how does this affect the equation? Justify your response.</p>
<p>A movie theatre can accommodate a maximum of 450 moviegoers per day. The theatre operators have determined that the profit per day, <code class='latex inline'>P</code>, is related to the ticket price, <code class='latex inline'>t</code>, by <code class='latex inline'>P=-30t^2+450t-790</code>. What ticket price will maximize the daily profit?</p>
<p>Martin wants to build an additional closet in a corner of his bedroom. Because the closet will be in a corner, only two new walls need to be built. The total length of the two new walls must be 12 m. Martin wants the length of the closet to be twice as long as the width, as shown in the diagram. Show your work.</p><p> Graph <code class='latex inline'>y = f(l)</code>.</p>
<p>The design of a new bridge can be modelled by the equation <code class='latex inline'>h = -0.0055d^2 + 25.2</code>, where <code class='latex inline'>h</code> is the height of the bridge, in metres, and <code class='latex inline'>d</code> is the length of the bridge, in metres.</p><p>a) Determine the length of the bridge, to the nearest tenth of a metre.</p><p>b) Determine the maximum height of the bridge above the ground, to the nearest tenth of a metre.</p><p>c) Determine the height of the bridge 42 m from the centre of the bridge, to the nearest tenth of a metre.</p>
<p>The parent council is planning the annual spaghetti supper to raise money for new school bleachers. Last year, the tickets sold for $1 1 each, and 400 people attended. This year the parent council has decided to raise the ticket price. They know that for every $1 increase in price, 20 fewer people will attend the supper.</p><p>a) What ticket price would maximize the revenue?</p><p>b) What is the maximum revenue?</p>
<p>The function <code class='latex inline'>P(x)=-30x^2+360x+785</code> models the profit, <code class='latex inline'>P(x)</code>, earned by a theatre owner on the basis of a ticket price, <code class='latex inline'>x</code>. Both the profit and ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?</p>
<ol> <li>Estimation a) Estimate the side length of a square that has the same area as a circle of radius <code class='latex inline'>\displaystyle 10 \mathrm{~cm} </code>. b) Check your estimate by finding the side length of the square, to the nearest hundredth of a centimetre.</li> </ol>
<p>A d1ver bounces off a 3-m springboard at an initial upward speed of 4 m/s.</p><p>a) Create a quadratic model for the height of the diver above the water.</p><p>b) After how many seconds does the diver enter the water? Round to the nearest hundredth of a second.</p><p>c) Over what time interval is the height of the diver greater than 3.5 m above the water? Round to the nearest hundredth of a second.</p>
<p>The area of a rectangular enclosure is given by the function <code class='latex inline'>A(w) = - 2w^2 + 48w</code>, where <code class='latex inline'>A(w)</code> is the area in square metres and <code class='latex inline'>w</code> is the width of the rectangle in metres.</p><p>A) What values of <code class='latex inline'>w</code> give an area of <code class='latex inline'>0</code>?</p><p>b) What is the maximum area of the enclosure?</p>
<p>Mario wants to install a wooden deck around a rectangular swimming pool. The function <code class='latex inline'>C(w)=120w^2+1800w</code> models the cost, where the cost, <code class='latex inline'>C(w)</code>, is in dollars and width, <code class='latex inline'>w</code>, is in metres. How wide will the deck be if he has $<code class='latex inline'>4080</code> to spend?</p>
<p>The profit function for a business is given by the equation <code class='latex inline'>P(x) = -4x^2 + 16x -7</code>, where <code class='latex inline'>x</code> is the number of items sold, in thousands, and <code class='latex inline'>P(x)</code> is dollars in thousands. Calculate the maximum profit and how many items must be sold to achieve it.</p>
<p>Beth wants to plant a garden at the back of her house. She has <code class='latex inline'>32</code> m of fencing. The area that can be enclosed is modelled by the function <code class='latex inline'>A(x)=-2x^2+32x</code>, where <code class='latex inline'>x</code> is the width of the garden in metres and <code class='latex inline'>A(x)</code> is the area in square metres. What is the maximum area that can be enclosed?</p><img src="/qimages/2196" />
<p>The area of a circle is given by the expression <code class='latex inline'>\pi(4x^2 + 36x + 81)</code>. What expression represents the diameter of this circle?</p>
<p>The area available for a swimming pool in a backyard is a square with side length 20 m. The square swimming pool is to be placed in the centre of the area. If the side length, in metres, of the swimming pool is <em>x</em>, then the area of the backyard remaining is given by the relation <code class='latex inline'>A= -x^2 +400</code>. </p><p>a) Graph the relation. </p><p>b) Find the intercepts. What do they represent?</p><p>c) For what values of <em>x</em> is the equation valid?</p>
<ol> <li>Canadian flag The Unity Flag is one of the largest Canadian flags. The length is twice the width, and the area is <code class='latex inline'>\displaystyle 167.2 \mathrm{~m}^{2} </code>. Find the dimensions of the Unity Flag, to the nearest tenth of a metre.</li> </ol>
<p>The population of a rural town can be modelled by the function <code class='latex inline'>P(x)=3x^2-102x+25\ 000</code>, where <code class='latex inline'>x</code> is the number of years since <code class='latex inline'>2000</code>. According to the model, when will the population be lowest?</p>
<p>A farming community collected data on the effect of different amounts of fertilizer, <code class='latex inline'>x</code>, in 100 kg/ha, on the yield of carrots, <code class='latex inline'>y</code>, in tonnes. The resulting quadratic regression model is <code class='latex inline'>y = -0.5x^2 + 1.4x + 0.1</code>. Determine the amount of fertilizer needed to produce the maximum yield.</p>
<p>A video tracking device recorded the height, <code class='latex inline'>h</code>, in metres, of a baseball after it was hit. The data collected can be modelled by the relation <code class='latex inline'>h=-5(t-2)^2+21</code>, where <code class='latex inline'>t</code> is the time in seconds after the ball was hit.</p> <ul> <li>At what time(s) was the baseball at a height of <code class='latex inline'>10</code> m?</li> </ul>
<p>A television screen is 40 cm high and 60 cm wide. The picture is compressed to 62.5% of its original area, leaving a uniform dark strip around the outside. What are the dimensions of the reduced picture?</p>
<p>Helen has 120 m of fencing. She plans to enclose a large rectangular garden and divide it into three equal parts.</p><img src="/qimages/21835" /><p>a) Write an area function in terms of <code class='latex inline'>x</code> to model the total area of the garden.</p><p>b) Determine the domain and the range for the area function.</p>
<p>A ball is thrown into the air from the roof of a building that is 25 m high. The ball reaches a maximum height of 45 m above the ground after 2 s and hits the ground 5 s after being thrown. </p><p><strong>a)</strong> Use the fact that the relation between time and the height of the ball is a quadratic relation to sketch an accurate graph of the relation. </p><p><strong>b)</strong> Carefully fold the graph along its axis of symmetry. Extend the short side of the parabola to match the long side. </p><p><strong>c)</strong> Where does the extended graph cross the time axis? </p><p><strong>d)</strong> What are the zeros of the relation? </p><p><strong>e)</strong> Determine the coordinates of the vertex. </p><p><strong>f)</strong> Determine an equation for the relation. </p><p><strong>g)</strong> What is the meaning of each zero? </p>
<p>The population of a city is modelled by <code class='latex inline'>\displaystyle P(t)=14 t^{2}+820 t+52000 </code>, where <code class='latex inline'>\displaystyle t </code> is time in years. Note: <code class='latex inline'>\displaystyle t=0 </code> corresponds to the year 2000 . According to the model, what will the population be in the year 2020? Here is Beverly&#39;s solution:</p><p>The population of a city is modelled by <code class='latex inline'>\displaystyle P(t)=14 t^{2}+820 t+52000 </code>, where <code class='latex inline'>\displaystyle t </code> is time in years. Note: <code class='latex inline'>\displaystyle t=0 </code> corresponds to the year 2000 . According to the model, what will the population be in the year 2020? Here is Beverly&#39;s solution:</p><p>Beverly&#39;s Solution</p><p><code class='latex inline'>\displaystyle \begin{array}{l}\begin{aligned} P(2020) &=14(2020)^{2}+820(2020)+52000 \\ &=57125600+1656400+52000 \\ &=58834000 & \text { I substituted } 2020 \text { into } \\ \text { the function for } t \\ \text { because that's the year } \\ \text { for which I want to } \end{aligned} \\ \text { The population will be } 58834000 . & \begin{array}{l}\text { know the population. } \\ \text { Then I solved. }\end{array}\end{array} </code></p><p>Are Beverly&#39;s solution and reasoning correct? Explain.</p>
<p>Think About a Plan The area of the rectangle shown is 80 square inches. What is the value of <code class='latex inline'>\displaystyle x ? </code></p> <ul> <li><p>How can you write an equation to represent 80 in terms of <code class='latex inline'>\displaystyle x ? </code></p></li> <li><p>How can you find the value of <code class='latex inline'>\displaystyle x </code> by completing the square?</p></li> </ul> <img src="/qimages/90076" />
<p>Solve by a table of values, a graphing calculator, and factoring: A computer software company models its profit with the function <code class='latex inline'>\displaystyle P(x)=-x^{2}+13 x-36 </code>, where <code class='latex inline'>\displaystyle x </code> is the number of games, in hundreds, that the company sells and <code class='latex inline'>\displaystyle P(x) </code> is the profit, in thousands of dollars.</p><p>a) How many games must the company sell to be profitable?</p><p>b) Which method do you prefer? Why?</p>
<p>When an object with a parachute is released to fall freely, its height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds is modelled by <code class='latex inline'>h=-0.5(g-r)t^2+k</code>, where <code class='latex inline'>g</code> is the acceleration due to gravity, <code class='latex inline'>r</code> is the resistance offered by the parachute, and <code class='latex inline'>k</code> is the height from which the object is dropped. On Earth, <code class='latex inline'>g=9.8</code>m/s<code class='latex inline'>^2</code>. The resistance offered by a single bed sheet is 0.6 m/s<code class='latex inline'>^2</code>, by a car tarp is 2.1 m/s<code class='latex inline'>^2</code>, and by a regular parachute is 8.9 m/s<code class='latex inline'>^2</code>.</p> <ul> <li>Is it possible to drop an object attached to the bed sheet and a similar object attached to a regular parachute and have them hit the ground at the same time? Describe how it would be possible and what the graphs of each might look like.</li> </ul>
<p>The diagram shows a square-shaped lawn with a <code class='latex inline'>\displaystyle 6 \mathrm{~m} </code> by <code class='latex inline'>\displaystyle 8 \mathrm{~m} </code> pool built inside.</p><img src="/qimages/155204" /><p>a) Find an equation expressing the area of the lawn in terms of the side length of the square. Sketch a graph for the equation.</p><p>b) Describe the transformation relative to a graph for the area of the lawn without the pool.</p>
<p>A stone is thrown from a bridge into a river. The height of the stone above the river at any time after it is released is modelled by the function <code class='latex inline'>\displaystyle h(t)=72-4.9 t^{2} . </code> The height of the stone, <code class='latex inline'>\displaystyle h(t) </code>, is measured in centimetres and time, <code class='latex inline'>\displaystyle t </code>, is measured in seconds.</p><p> If <code class='latex inline'>\displaystyle h(3)=27.9 </code>, explain what you know about the stone&#39;s position.</p>
<ol> <li>A flaming arrow is fired upward from the deck of a ship to mark the beginning of an evening of entertainment and</li> </ol> <p>celebration. The flaming arrow hits the</p><p>water. The height, <code class='latex inline'>\displaystyle h </code>, in metres, of the arrow above the water <code class='latex inline'>\displaystyle t </code> seconds after it is fired can be modelled by the quadratic function <code class='latex inline'>\displaystyle h(t)=-4.9 t^{2}+98 t+8 . </code> a) Determine the maximum height of</p><p>the arrow.</p><p>b) How long does it take the arrow to</p><p>reach its maximum height?</p><p>c) When does the arrow hit the water?</p><p>d) How high is the deck of the ship</p><p>above the water?</p>
<p>The height of water, <code class='latex inline'>\displaystyle h(t) </code>, in metres, from a garden hose is given by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+15 t </code>, where <code class='latex inline'>\displaystyle t </code> is time in seconds. Express the function in factored form, and then use the zeros to determine the maximum height reached by the water.</p>
<p>The height of a soccer ball kicked in the air is given by the quadratic equation <code class='latex inline'>h(t)=-4.9t(t-2.1)^2+23</code>, where time, <code class='latex inline'>t</code>, is in seconds and height, <code class='latex inline'>h(t)</code>, is in metres.</p><p>Is the ball still in the air after <code class='latex inline'>6</code> s? Explain.</p>
<p>The outward power, <code class='latex inline'>\displaystyle P </code>, in watts, of a <code class='latex inline'>\displaystyle 120-V </code> electric generator is given by the relation <code class='latex inline'>\displaystyle P=120 I-5 I^{2} </code>, where <code class='latex inline'>\displaystyle I </code> is the current, in amperes (A). For what value(s) of <code class='latex inline'>\displaystyle I </code> is the outward power <code class='latex inline'>\displaystyle 500 \mathrm{~W} </code> ?</p>
<p>Forty metres of fencing are available to enclose a rectangular pen. When the length of the pen is l metres, the area of the pen, in square metres, is expressed as <code class='latex inline'>\displaystyle 20 l - l^2 </code> What is the area of the pen for each value of <code class='latex inline'>l</code>?</p><p>a) 4 m</p><p>b )10 m</p><p>c) 13 m</p>
<p>The path of a firework is modelled using the equation <code class='latex inline'>h = -5d^2 + 20d+ 1</code>, where <code class='latex inline'>h</code> is the height, in metres, above the ground and <code class='latex inline'>d</code> is the horizontal distance, in metres. What is the maximum height of the firework?</p>
<img src="/qimages/64268" /> <ol> <li>Brooklyn Bridge The Brooklyn Bridge in New York City is a suspension bridge that crosses the East River and connects Brooklyn to the island of Manhattan. If the origin is placed at the top of one of the cablesupport towers, as shown, the shape of a cable that supports the main span can be modelled by the equation</li> </ol> <p><code class='latex inline'>\displaystyle h=0.0008 d^{2}-0.384 d </code></p><p>where <code class='latex inline'>\displaystyle h </code> metres represents the height and <code class='latex inline'>\displaystyle d </code> metres represents the horizontal distance.</p><p>a) What is the vertical distance from the top of a support tower to the lowest point on a cable, to the nearest metre?</p><p>b) What is the length of the main span?</p><p>c) At a horizontal distance of <code class='latex inline'>\displaystyle 50 \mathrm{~m} </code> from one end of the cable, how far is the cable below the top of the support towers, to the nearest metre?</p>
<p>A video tracking device recorded the height, <code class='latex inline'>h</code>, in metres, of a baseball after it was hit. The data collected can be modelled by the relation <code class='latex inline'>h=-5(t-2)^2+21</code>, where <code class='latex inline'>t</code> is the time in seconds after the ball was hit.</p> <ul> <li>What was the maximum height reached by the baseball?</li> </ul>
<p>For the relation, explain what each coordinate of the vertex represents and what the zeros represent.</p> <ul> <li>a relation that models the height, <code class='latex inline'>h</code>, of a ball that has been kicked from the ground after time <code class='latex inline'>t</code></li> </ul>
<ol> <li>Soccer The equation shows the height of a soccer ball, <code class='latex inline'>\displaystyle h </code> metres, as a function of the horizontal distance, <code class='latex inline'>\displaystyle d </code> metres, the ball travels until it first hits the ground.</li> </ol> <p><code class='latex inline'>\displaystyle h=-0.025(d-20)^{2}+10 </code> a) What is the maximum height of the ball? b) What is the horizontal distance of the ball from the kicker when it reaches its maximum height?</p><p>c) How far does the ball travel horizontally from when it is kicked until it hits the ground?</p><p>d) What is the height of the ball when it is <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> horizontally from the kicker?</p><p>e) Communication Would an opposing player positioned under the path of the ball <code class='latex inline'>\displaystyle 34 \mathrm{~m} </code> from the kicker be able to head the ball? Explain. f) If the origin were placed at the vertex of the parabola, what would be the equation of the curve?</p>
<p>A rectangular swimming pool has a row of water fountains along each of its two longer sides. The two rows of fountains are 10 m apart. Each fountain sprays an identical parabolic—shaped stream of water a total horizontal distance of 8 m toward the opposite Side. Looking from one end of the pool, the streams of water from the two sides cross each other in the middle of the pool at a height of 3 m.</p><p><strong>a)</strong> Determine an equation that represents a stream of water from the left side and another equation that represents a stream of water from the right side. Graph both equations on the same set of axes.</p><p><strong>b)</strong> Determine the maximum height of the water.</p><img src="/qimages/1539" />
<p>Determine the break-even quantities for each profit function, where <code class='latex inline'>x</code> is the number sold, in thousands.</p><p><code class='latex inline'> \displaystyle P(x) = -2x^2 +18x - 40 </code></p>
<p>The area, <code class='latex inline'>A</code>, of a square is related to its perimeter, <code class='latex inline'>P</code>, by the formula <code class='latex inline'>A=\displaystyle{\frac{P^2}{16}}</code></p> <ul> <li>Rearrange this formula to express <code class='latex inline'>P</code> in terms of <code class='latex inline'>A</code>.</li> </ul>
<p>A ball is thrown into water from a cliff that is <code class='latex inline'>\displaystyle 175 \mathrm{~m} </code> high. The height of the ball above the water after it is thrown is modelled by the function <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+10 t+175 </code>, where <code class='latex inline'>\displaystyle b(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is time in seconds. a) When will the ball reach the water below the cliff? b) When will the ball reach a ledge that is <code class='latex inline'>\displaystyle 100 \mathrm{~m} </code> above the water?</p>
<p>MOVIE THEATER A company plans to build a large multiplex theater. The financial analyst told her manager that the profit function for their theater was <code class='latex inline'>\displaystyle P(x)=-x^{2}+48 x-512 </code>, where <code class='latex inline'>\displaystyle x </code> is the number of movie screens, and <code class='latex inline'>\displaystyle P(x) </code> is the profit earned in thousands of dollars. Determine the range of production of movie screens that will guarantee that the company will not lose money.</p>
<p>The monthly profit, <code class='latex inline'>\displaystyle P(x) </code>, of a sportswear company, in thousands of dollars, is</p><p>represented by the quadratic function</p><p><code class='latex inline'>\displaystyle P(x)=-3 x^{2}+18 x-2 </code>, where <code class='latex inline'>\displaystyle x </code> is the amount spent on advertising, in thousands of dollars.</p><p>a) Determine the company&#39;s maximum</p><p>monthly profit.</p><p>b) Determine the amount spent on</p><p>advertising to achieve the maximum</p><p>profit.</p>
<p>a) Write an algebraic expression to represent the surface area of the rectangular prism. Expand and simplify.</p><img src="/qimages/9416" /><p>b) If <code class='latex inline'>x</code> represents 5 cm, what is the surface area?</p>
<p>The height of a soccer ball kicked in the air is given by the quadratic equation <code class='latex inline'>h(t)=-4.9t(t-2.1)^2+23</code>, where time, <code class='latex inline'>t</code>, is in seconds and height, <code class='latex inline'>h(t)</code>, is in metres.</p><p> What was the height of the ball when it was kicked?</p>
<ol> <li>Measurement Find the dimensions of both squares in each diagram. The area of the shaded region is given in each case.</li> </ol> <img src="/qimages/64502" />
<p>Baking Your local bakery sells more bagels when it reduces prices, but then its profit changes. The function <code class='latex inline'>\displaystyle y=-1000 x^{2}+1100 x-2.5 </code> models the bakery&#39;s daily profit in dollars, from selling bagels, where <code class='latex inline'>\displaystyle x </code> is the price of a bagel in dollars. What&#39;s the highest price the bakery can charge, in dollars, and make a profit of at least <code class='latex inline'>\displaystyle \$ 200 ? </code></p>
<p>The function <code class='latex inline'>\displaystyle d(s)=0.0056 s^{2}+0.14 s </code> models the stopping distance of a car, <code class='latex inline'>\displaystyle d(s) </code>, in metres, and the speed, <code class='latex inline'>\displaystyle s </code>, in kilometres per hour. What is the speed when the stopping distance is <code class='latex inline'>\displaystyle 7 \mathrm{~m} </code> ? Use a graph to solve.</p>
<p>For the relation, explain what each coordinate of the vertex represents and what the zeros represent.</p> <ul> <li>a relation that models the height, <code class='latex inline'>h</code>, of a ball when it is a distance of <code class='latex inline'>x</code> metres from where it was thrown from a second-floor balcony</li> </ul>
<p>A bowling alley has a <code class='latex inline'>\$5</code> cover charge on Friday nights. The manager is considering increasing the cover charge in <code class='latex inline'>50</code>¢ increments. The revenue is modelled by the function <code class='latex inline'>R(x)=-12.5x^2+75x+2000</code>, where revenue <code class='latex inline'>R(x)</code> is in dollars and <code class='latex inline'>x</code> is the number of <code class='latex inline'>50</code>¢ increments.</p><p>What will the cover charge be if the revenue is <code class='latex inline'>\$2000</code>?</p>
<p>The length of one leg of a right triangle is 17 cm more than that of the other leg. The length of the hypotenuse is 4 cm more than triple that of the shorter leg. Find the lengths of each of the three sides.</p><img src="/qimages/21784" />
<ol> <li>Cropping a photograph If part of a photograph is used to fill an available space in a book or magazine, the photograph is said to be cropped. A photograph that was originally <code class='latex inline'>\displaystyle 15 \mathrm{~cm} </code> by <code class='latex inline'>\displaystyle 10 \mathrm{~cm} </code> is cropped by removing the same width from the top and the left side. Cropping reduces the area by <code class='latex inline'>\displaystyle 46 \mathrm{~cm}^{2} </code>. What are the dimensions of the cropped photograph?</li> </ol>
<p>The parabolic cross section of an arch in front of a museum is modelled by the relation <code class='latex inline'>h=-d^2+9</code>, where <code class='latex inline'>h</code> is the height, in metres, above the ground and <code class='latex inline'>d</code> is the horizontal distance, in metres, from the centre of the arch.</p><p>a) How wide and how tall is the arch?</p><p>b) Sketch a graph to represent the cross section.</p><p>c) For what values of <code class='latex inline'>d</code> is the relation valid? Explain.</p>
<p>The fuse of a three-break firework rocket is programmed to ignite three times with 2-second intervals between the ignitions. When the rocket is shot vertically in the air, its height <code class='latex inline'>\displaystyle h </code> in feet after <code class='latex inline'>\displaystyle t </code> seconds is given by the formula <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+70 t </code>. At how many seconds after the shot should the firework technician set the timer of the first ignition to make the second ignition occur when the rocket is at its highest point? <code class='latex inline'>\displaystyle \begin{array}{lll}\text { A } 3 & \text { B } 9 & \text { C } 5\end{array} </code></p>
<p>The population of a town is modelled by the function <code class='latex inline'>P(t)=6t^2+110t+4000</code>, where <code class='latex inline'>P(t)</code> is the population and <code class='latex inline'>t</code> is the time in years since <code class='latex inline'>2000</code>.</p><p>When will the population be <code class='latex inline'>6000</code>?</p>
<ol> <li>Lidless box A rectangular piece of tin <code class='latex inline'>\displaystyle 50 \mathrm{~cm} </code> by <code class='latex inline'>\displaystyle 40 \mathrm{~cm} </code> is made into a lidless box of base area <code class='latex inline'>\displaystyle 875 \mathrm{~cm}^{2} </code> by cutting squares of equal sizes from the corners and bending up the sides. Find</li> </ol> <p>a) the side length of each removed square b) the volume of the box</p>
<p>Sara has designed a fishpond in the shape of a right triangle with two sides of length <code class='latex inline'>a</code> and <code class='latex inline'>b</code> and hypotenuse of length <code class='latex inline'>c</code>.</p><p>The hypotenuse is <code class='latex inline'>3m</code> longer than <code class='latex inline'>b</code>, and the sum of the lengths of the hypotenuse and <code class='latex inline'>b</code> is <code class='latex inline'>11 m</code>. What are the lengths of the sides of the pond?</p>
<p>Kool Klothes has determined that the revenue function for selling <code class='latex inline'>x</code> thousand pairs of shorts is <code class='latex inline'>R(x) = - 5x^2 + 21x</code>. The cost function <code class='latex inline'>C(x) = 2x + 10</code> is the cost of producing the shorts.</p><p>a) Write a profit function.</p><p>b) How many pairs of shorts must the company sell in order to break even?</p>
<p>A stone is thrown from a bridge into a river. The height of the stone above the river at any time after it is released is modelled by the function <code class='latex inline'>\displaystyle h(t)=72-4.9 t^{2} . </code> The height of the stone, <code class='latex inline'>\displaystyle h(t) </code>, is measured in centimetres and time, <code class='latex inline'>\displaystyle t </code>, is measured in seconds.</p><p>a) Evaluate <code class='latex inline'>\displaystyle h(0) </code>. What does it represent?</p><p>b) Evaluate <code class='latex inline'>\displaystyle h(2.5) </code>. What does it represent?</p><p>c) If <code class='latex inline'>\displaystyle h(3)=27.9 </code>, explain what you know about the stone&#39;s position.</p>
<p> Find the vertex of the following parabolas and state which parabola(with its vertex at the origin) it is congruent to. </p><p><code class='latex inline'>\displaystyle y = 3x^2 - 24x + 40 </code></p>
<p>The stainless-steel Gateway Arch in St. Louis, Missouri, is almost parabolic in shape. It is <code class='latex inline'>\displaystyle 192 \mathrm{~m} </code> from the base of the left leg to the base of the right leg. The arch is <code class='latex inline'>\displaystyle 192 \mathrm{~m} </code> high. Determine a function, in standard form, that models the shape of the arch.</p>
<p>Find two consecutive even integers with a product of 624</p>
<p>The formula for the surface area of a cylinder is <code class='latex inline'>SA=2\pi r^2+2\pi rh</code>. u A cylinder has a height of <code class='latex inline'>10</code> units and a radius of r units. Determine a factored expression for its total surface area. </p>
<p>A bowling alley has a <code class='latex inline'>\$5</code> cover charge on Friday nights. The manager is considering increasing the cover charge in <code class='latex inline'>50</code>¢ increments. The revenue is modelled by the function <code class='latex inline'>R(x)=-12.5x^2+75x+2000</code>, where revenue <code class='latex inline'>R(x)</code> is in dollars and <code class='latex inline'>x</code> is the number of <code class='latex inline'>50</code>¢ increments.</p><p>What cover charge will yield the maximum revenue?</p>
<ol> <li>Measurement The hypotenuse of a right triangle measures <code class='latex inline'>\displaystyle 20 \mathrm{~cm} </code>. The sum of the lengths of the other two sides is <code class='latex inline'>\displaystyle 28 \mathrm{~cm} </code>. Find the lengths of these two sides.</li> </ol>
<p>The area, <code class='latex inline'>A</code>, of each figure is given. Determine the unknown measurement.</p><p><code class='latex inline'>A=10m^2-20m+20</code></p><img src="/qimages/2220" />
<p>The world production of gold from 1970 to 1990 can be modelled by <code class='latex inline'>G=1492-76t+5.2t^2</code>, where <code class='latex inline'>G</code> is the number of tonnes of gold and <code class='latex inline'>t</code> is the number ofyears since 1970 (<code class='latex inline'>t=0</code> for 1970, <code class='latex inline'>t=1</code> for 1971, and so on).</p> <ul> <li>During this period, when was the minimum amount of gold mined?</li> </ul>
<p>The cost, in dollars, of operating an appliance per day is given by the formula <code class='latex inline'>C=2t^2-24t+150</code>, where <code class='latex inline'>t</code> is the time, in months, the appliance is running. What is the minimum cost of running the appliance?</p>
<p>A caterer in quoting the charge for producing a dinner proposes the following terms For a group of 60 people. he will charge $30 per person. For every extra 10 people he will lower the price by $1.50 per person for the whole group. What size group ’ does the caterer want to maximize his income?</p>
<p>The height of a soccer ball kicked in the air is given by the quadratic equation <code class='latex inline'>h(t)=-4.9t(t-2.1)^2+23</code>, where time, <code class='latex inline'>t</code>, is in seconds and height, <code class='latex inline'>h(t)</code>, is in metres.</p><p>When is the ball at a height of <code class='latex inline'>10</code> m?</p>
<p>A rectangular box is 20 cm high and twice as long as it is wide. If it has a surface area of 1600 <code class='latex inline'>cm^2</code>, what is its volume?</p>
<p>A rock is thrown down from a cliff that is 180 m high. The function <code class='latex inline'>h(t) = -5t^2 -10t + 180</code> gives the approximate height of the rock above the water, where <code class='latex inline'>h(t)</code> is the height in metres and t is the time in seconds. When will the rock reach a ledge that is 105 m above the water?</p>
<p>Van dives off a 4-m springboard. His height, <code class='latex inline'>h</code>, in metres, above the surface of the water is defined by the relation <code class='latex inline'>h = -d^2 + 3d + 4</code>, where <code class='latex inline'>d</code> is his horizontal distance, in metres, from the end of the board.</p><p>a) Determine the zeros of the relation.</p><p>b) Sketch a graph of the relation.</p><p>c) For what values of d is the relation valid?</p><p>d) What is Van’s horizontal distance from the board when he enters the water?</p><p>e) What is Van’s maximum height above the water?</p>
<p>An arch of a highway overpass is in the shape of a parabola. The arch spans a distance of <code class='latex inline'>12</code> m from one side of the road to the other. The height of the arch is <code class='latex inline'>8</code> m at a horizontal distance of <code class='latex inline'>2</code> m from each side of the arch.</p><p><strong>(a)</strong> Sketch the quadratic function if the vertex of the parabola is on the y-axis and the road is along the x-axis.</p><p><strong>(b)</strong> Use this information to determine the function that models the arch.</p><p><strong>(c)</strong> Find the maximum height of the arch to the nearest tenth of a metre.</p>
<p>Think About a Plan Tran&#39;s truck gets very poor gas mileage. If Tran pays <code class='latex inline'>\displaystyle \$ 84 </code> to fill his truck with gas and is able to drive <code class='latex inline'>\displaystyle m </code> miles on a full tank, what expression shows his gas cost per mile?</p> <ul> <li><p>What operation does &quot;per&quot; indicate?</p></li> <li><p>Check your expression by substituting 200 miles for <code class='latex inline'>\displaystyle m </code>. Does your answer make sense?</p></li> </ul>
<p>Write and simplify an expression to represent the area of each figure. </p><img src="/qimages/155603" />
<p>An arch of a highway overpass is in the shape of a parabola. The arch spans a distance of <code class='latex inline'>12</code> m from one side of the road to the other. The height of the arch is <code class='latex inline'>8</code> m at a horizontal distance of <code class='latex inline'>2</code> m from each side of the arch.</p><p>From above, instead of having the vertex on the y-axis, put one side of the archway at the origin of the grid. You will get a different wheat ion because the eros are now at <code class='latex inline'>0</code> and <code class='latex inline'>12</code>, rather than at <code class='latex inline'>-6</code> and <code class='latex inline'>6</code>.</p><p><strong>(a)</strong> Find the equation of the quadratic function for this position.</p><p><strong>(b)</strong> Find the maximum height of the overpass and compare the result to the height calculated if the y-axis was the center.</p>
<p>The flight of an aircraft from Toronto to Montreal can be modelled by the relation <code class='latex inline'>h = -2.5t^2 + 200t</code>, where <code class='latex inline'>t</code> is the time, in minutes, and <code class='latex inline'>h</code> is the height, in metres.</p><p>a) How long does it take to fly from Toronto to Montreal?</p><p>b) What is the maximum height of the aircraft? At what time does the aircraft reach this height?</p>
<p>Write an expression for the area of each shape. Expand and simplify. </p><img src="/qimages/2214" />
<p>The period of a pendulum is the time the pendulum takes to swing back and forth. The function <code class='latex inline'>\displaystyle L=0.81 t^{2} </code> relates the length <code class='latex inline'>\displaystyle L </code> in feet of a pendulum to the time <code class='latex inline'>\displaystyle t </code> in seconds that it takes to swing back and forth. A convention center has a pendulum that is 90 feet long. Find the period.</p>
<p>Which pair of numbers that add to 10 will multiply to give the greatest product?</p>
<p>When a 3-m springboard diver leaves the diving board, her height above the water depends on the time since she left the board. When the time is <code class='latex inline'>t</code> seconds, the diver’s height above the water, in metres, is expressed as <code class='latex inline'>3 + 8,8t -4.9t^2</code>.</p><p>Find the height of the diver after each time.</p><p>a) 0.5 s</p><p>b) 1 s</p><p>c) 1.5 s</p>
<p>A movie theatre can accommodate a maximum of 350 moviegoers per day. The theatre operators have been changing th admission price to find out how price affects ticket sales and profit. Currently, they charge <code class='latex inline'>\$11</code> a person and sell about 300 tickets per day. After reviewing their data, the theatre operators discovered that they could express the relation between profit, P, and the number of <code class='latex inline'>\$1</code> price increase, <code class='latex inline'>x</code>, as <code class='latex inline'>P = 20(15 - x)(11+ x)</code></p><p><strong>a)</strong> Determine the vertex form of the profit equation.</p><p><strong>b)</strong> What ticket price results in the maximum profit? What is the maximum profit? About how many tickets will be sold at this price?</p>
<p>The traffic safety bureau receives data regarding acceleration of a prototype electric sports car. It can accelerate from 0 to 100 km/h in about 4 s. Its position, d, in metres, at any time t, in seconds, is given by <code class='latex inline'>d(t)=3.5t^2</code>. Mathew is comparing the prototype to a hybrid electric car, which has its position given by <code class='latex inline'>d(t)=1.4t^2</code>. </p><p><strong>(a)</strong> In a race between the two cars, the hybrid is given a head start. Where would the hybrid have to start so that after 4 s of acceleration, both cars are in the same position?</p><p><strong>(b)</strong> Verify your solution by graphing in part (a).</p>
<p>The number of glasses of pink lemonade, n, sold by the Lemonade Dreams Cafe on a given day is modelled by <code class='latex inline'>n = 300 -50p</code>, where <code class='latex inline'>p</code> is the price, in dollars.</p><p>a) Solve this equation for <code class='latex inline'>p</code>. </p><p>b) The revenue generated by lemonade sales is <code class='latex inline'>R = 2np</code>. Substitute your expression for <code class='latex inline'>p</code> from part a), and expand to obtain an expression for the daily pink lemonade revenue.</p>
<p>The sum of two numbers is 26 , and the sum of their squares is a minimum.</p><p>a) Determine the numbers.</p><p>b) What is the minimum sum of their</p><p>squares?</p>
<p>a) Sketch two different rectangles with an area of <code class='latex inline'>4x^2+8x</code>.</p><p>b) List the dimensions of each rectangle.</p>
<p>The height, <code class='latex inline'>\displaystyle h </code>, in metres, of an object <code class='latex inline'>\displaystyle t </code> seconds after it is dropped is <code class='latex inline'>\displaystyle h=-0.5 g t^{2}+k </code>, where <code class='latex inline'>\displaystyle g </code> is the acceleration due to gravity and <code class='latex inline'>\displaystyle k </code> is the height from which the object is released. If an object is released from a height of <code class='latex inline'>\displaystyle 400 \mathrm{~m} </code>, how much longer does it take to fall to a height of <code class='latex inline'>\displaystyle 75 \mathrm{~m} </code> on the Moon compared with falling to the same height on Earth? The acceleration due to gravity is <code class='latex inline'>\displaystyle 9.8 \mathrm{~m} / \mathrm{s}^{2} </code> on Earth and <code class='latex inline'>\displaystyle 1.6 \mathrm{~m} / \mathrm{s}^{2} </code> on the Moon.</p>
<ol> <li>Natural bridge Sipapu Natural Bridge is in Utah. Find the horizontal distance, in metres, across this natural arch at the base by solving the equation <code class='latex inline'>\displaystyle -0.04 x^{2}+3.28 x=0 </code>.</li> </ol>
<p>Dianne dove from the 10-m diving board. Her height h, in metres, above the water when she is x metres away from the end of the board is given by <code class='latex inline'>h = -(x - 1]^2 + 11</code>.</p><p>a) Sketch a graph of her dive.</p><p>b) What was her maximum height above the water?</p><p>c) What horizontal distance had she travelled when she entered the water? Answer to the nearest tenth of a metre.</p>
<p>Martin wants to build an additional closet in a corner of his bedroom. Because the closet will be in a corner, only two new walls need to be built. The total length of the two new walls must be <code class='latex inline'>12 m</code>. Martin wants the length of the closet to be twice as long as the width, as shown in the diagram. </p><p>Find the desired length and width. Show your work.</p>
<img src="/qimages/17727" /><p>Calculate the value of <code class='latex inline'>\displaystyle x . </code></p>
<p>A small rocket is launched. It reaches a maximum height of <code class='latex inline'>\displaystyle 120 \mathrm{~m} </code> and lands <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> from the launching pad. Assume the rocket follows a parabolic path. Write the equation that describes its height, <code class='latex inline'>\displaystyle h </code> metres, as a function of its horizontal distance, <code class='latex inline'>\displaystyle x </code> metres, from the launching pad.</p>
<img src="/qimages/63951" /> <ol> <li>Measurement The 25 by 16 rectangle contains a square of side length <code class='latex inline'>\displaystyle s </code>. The sides of the square are parallel to the sides of the rectangle. a) Write the area of the shaded region, <code class='latex inline'>\displaystyle A </code>, as a function of <code class='latex inline'>\displaystyle s </code>. b) If no part of the square can be outside the rectangle, what is the maximum possible value of <code class='latex inline'>\displaystyle s </code> ? c) Graph <code class='latex inline'>\displaystyle A </code> versus <code class='latex inline'>\displaystyle s </code>. d) State the domain and range of the function.</li> </ol>
<p>Write and simplify an expression to represent the area of the shaded region. <code class='latex inline'>\displaystyle 3 x+4 \mid x-1 </code></p><img src="/qimages/155604" />
<p>The sum of a number and its reciprocal is 6.41. What is the number?</p>
<p>An inflatable raft is dropped from hovering helicopter to boat in distress below. The height of the raft above the water, <code class='latex inline'>y</code>, in metres, is approximated by the equation <code class='latex inline'>y=500-5x^2</code>, where <code class='latex inline'>x</code> is the time in seconds since the raft was dropped.</p> <ul> <li>What is the height of the raft above the water 6 s after it is dropped?</li> </ul>
<p>A lifeguard has <code class='latex inline'>\displaystyle 40 \mathrm{~m} </code> of rope to enclose a rectangular swimming area at a small</p><p>lake. One side of the rectangle is a straight sandy beach. What are the dimensions of</p><p>the largest swimming area that she can</p><p>enclose?</p>
<p>A local club alternates between bands and booking DJs. By tracking receipts over a period of time, the owner of the club determined that her profit from a live band depended on the ticket price. Her profit, <code class='latex inline'>P</code>, can be modelled using <code class='latex inline'>P = -15x^2 + 600x + 50</code>, where <code class='latex inline'>x</code> represents the ticket price in dollars.</p><p>a) Sketch the graph of the relation to help the owner understand this profit model.</p><p>b) Determine the maximum profit and the ticket price she should charge to achieve the maximum profit.</p>
<p>A water balloon was launched from catapult. The table shows the data collected during the flight of the balloon using stop-motion photography.</p><img src="/qimages/779" /> <ul> <li>Use the equation <code class='latex inline'>y = -0.04x(x - 54)</code> to determine the height of the balloon when its horizontal distance was 40 m.</li> </ul>
<p>Geometry The table shows some possible dimensions of rectangles with a perimeter of 100 units. Copy and complete the table.</p><p>a. Plot the points (width, area). Find a model for the data set.</p><p>b. What is another point in the data set? Use it to verify your model.</p><p>c. What is a reasonable domain for this function? Explain.</p><p>d. Find the maximum possible area. What dimensions yield this area?</p><p>e. Find a function for area in terms of width without using the table. Do you get the same model as in part (a)? Explain.</p><img src="/qimages/90088" />
<p>A number of students charter a bus to go to a school football game at a total cost of $80. Eight of the students are ill and cannot go. Each of the remaining students then has to pay an extra 50¢. How many students go on the bus?</p>
<p>The path of a football is modelled by the relation <code class='latex inline'> h = -\dfrac{1}{4} (d-12)^2 + 36</code>, where <em>d</em> is the horizontal distance, in metres, after it was kicked, and <em>h</em> is the height, in metres, above the ground. </p><p>a) Sketch the path of the football. </p><p>b) What is the horizontal distance when this occurs?</p><p>c) What is the horizontal distance when this occurs?</p><p>d) What is the height of the football at a horizontal distance of 10 m?</p><p>e) Find another horizontal distance where the height is the same as in part d). </p>
<p>The population of a town is modelled by the function <code class='latex inline'>P(t)=6t^2+110t+4000</code>, where <code class='latex inline'>P(t)</code> is the population and <code class='latex inline'>t</code> is the time in years since <code class='latex inline'>2000</code>.</p><p>Will the population ever be <code class='latex inline'>0</code>? Explain your answer.</p>
<p>The manager of a hardware store knows that the weekly revenue function for batteries sold can be modelled with <code class='latex inline'>R(x)= -x^2 + 10x + 30 000</code>, where both the revenue, <code class='latex inline'>R(x)</code>, and the cost, <code class='latex inline'>x</code>, of a package of batteries are in dollars. According to the model, what is the maximum revenue the store will earn?</p>
<p>The expression <code class='latex inline'>16t^2</code> models the distance in feet that an object falls during the first <code class='latex inline'>t</code> seconds after being dropped. What is the distance the object falls during each time?</p><p>0.25 second</p>
<p>A ball is tossed upward from a cliff that is <code class='latex inline'>\displaystyle 40 \mathrm{~m} </code> above water. The height of the ball above the water is modelled by <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+10 t+40 </code>, where <code class='latex inline'>\displaystyle b(t) </code> is the height in metres and <code class='latex inline'>\displaystyle t </code> is the time in seconds. Use a graph to answer the following questions. a) What is the maximum height reached by the ball? b) When will the ball hit the water?</p>
<p>The world production of gold from 1970 to 1990 can be modelled by <code class='latex inline'>G=1492-76t+5.2t^2</code>, where <code class='latex inline'>G</code> is the number of tonnes of gold and <code class='latex inline'>t</code> is the number ofyears since 1970 (<code class='latex inline'>t=0</code> for 1970, <code class='latex inline'>t=1</code> for 1971, and so on).</p> <ul> <li>What was the least amount of gold mined in one year?</li> </ul>
<p>Write an algebraic expression to represent the area of the figure. Expand and simplify.</p><img src="/qimages/6057" />
<p>The world production of gold from 1970 to 1990 can be modelled by <code class='latex inline'>G=1492-76t+5.2t^2</code>, where <code class='latex inline'>G</code> is the number of tonnes of gold and <code class='latex inline'>t</code> is the number ofyears since 1970 (<code class='latex inline'>t=0</code> for 1970, <code class='latex inline'>t=1</code> for 1971, and so on).</p> <ul> <li>How much gold was mined in 1985?</li> </ul>
<p>A theatre company’s profit, <code class='latex inline'>P(x)</code>, on a production is modelled by <code class='latex inline'>P(x) = -60x^2 + 1800x + 16 500</code>, where <code class='latex inline'>x</code> is the cost of a ticket in dollars. According to the model, what should the company charge per ticket to make the maximum profit?</p>
<p>The population of a city, <code class='latex inline'>\displaystyle P(t) </code>, is given by the function <code class='latex inline'>\displaystyle P(t)=14 t^{2}+820 t+42000 </code>, where <code class='latex inline'>\displaystyle t </code> is time in years. Note: <code class='latex inline'>\displaystyle t=0 </code> corresponds to the year 2000 .</p><p>a) When will the population reach 56224 ? Provide your reasoning.</p><p>b) What will the population be in 2035 ? Provide your reasoning.</p>
<p>Babe Ruth, a baseball player, hits a &quot;major league pop-up&quot; so that the height of the ball, in metres, is modelled by the function <code class='latex inline'>\displaystyle h(t)=1+30 t-5 t^{2} </code>, where <code class='latex inline'>\displaystyle t </code> is time in seconds.</p><p>a) Evaluate the function for each of the given times to complete the table of values below.</p><p>b) Graph the function.</p><p>c) When does the ball reach its maximum height?</p><p>d) What is the ball&#39;s maximum height?</p><p>e) How long does it take for the ball to hit the ground?</p><p><code class='latex inline'>\displaystyle \begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Time (\mathbf{s}) & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 & 4.5 & 5.0 & 5.5 & 6.0 & 6.5 & 7.0 & 7.5 & 8.0 \\ \hline Height (\mathbf{m}) & & & & & & & & & & & & & & & & & \\ \hline \end{array} </code></p>
<p>a) Write three polynomials whose terms have a greatest common factor of <code class='latex inline'>4x^3 y</code>.</p><p>b) Factor each polynomial you wrote in part a)</p>
<p>The height in metres of a projectile launched from the top of a building is given by <code class='latex inline'>\displaystyle h(t)=-5 t^{2}+20 t+15 </code>, where <code class='latex inline'>\displaystyle t </code> is the time in seconds since it was launched.</p><p>a) How high was the projectile at the moment of launch? b) At what time does the projectile hit the ground? c) What is the average rate of change in height from the time the object was launched until the time it hit the ground?</p>
<p>Accountants for the HiTech Shoe Company have determined that the quadratic relation <code class='latex inline'>P = -2x^2 + 24x - 54</code> models the company’s profit for the next quarter. In this relation, P represents the profit (in $100 0005) and x represents the number of pairs of shoes sold (in 100 0005).</p><p>a) Express the equation in factored form.</p><p>b) What are the zeros of the relation? What do they represent in this context?</p><p>c) Determine the number of pairs of shoes that the company must sell to maximize its profit. How much would the maximum profit be?</p>
<p>Two numbers have a difference of 8 .</p><p>a) What is the maximum product of</p><p>these numbers?</p><p>b) What are the numbers that produce</p><p>the maximum product?</p>
<p>The relation <code class='latex inline'>I = 0.045s^2</code> can be used to calculate the length, <code class='latex inline'>I</code>, in metres, of the skid mark for a car travelling at a speed, <code class='latex inline'>s</code>, in kilometres per hour, on dry pavement before braking.</p><p><em>(a)</em> What is the length of the skid mark for a car travelling at 50 km/h? 100 km/h?</p><p><strong>answer b) only</strong></p><p><strong>(b)</strong> How do the results in part a) compare?</p>
<p>When an object with a parachute is released to fall freely, its height, <code class='latex inline'>h</code>, in metres, after <code class='latex inline'>t</code> seconds is modelled by <code class='latex inline'>h=-0.5(g-r)t^2+k</code>, where <code class='latex inline'>g</code> is the acceleration due to gravity, <code class='latex inline'>r</code> is the resistance offered by the parachute, and <code class='latex inline'>k</code> is the height from which the object is dropped. On Earth, <code class='latex inline'>g=9.8</code>m/s<code class='latex inline'>^2</code>. The resistance offered by a single bed sheet is 0.6 m/s<code class='latex inline'>^2</code>, by a car tarp is 2.1 m/s<code class='latex inline'>^2</code>, and by a regular parachute is 8.9 m/s<code class='latex inline'>^2</code>.</p> <ul> <li>Describe how the graphs will differ for objects dropped from a height of 100 m using each of the three types of parachutes.</li> </ul>
<p>Acceleration due to gravity, <code class='latex inline'>a</code>, varies from planet to planet. The distance of the object from the drop location after <code class='latex inline'>t</code> seconds is given by <code class='latex inline'>d(t) = \frac{a}{2}t^2</code>. The table shows the estimated acceleration due to gravity for different planets.</p><p>a) State the base function, <code class='latex inline'>f(t),</code> for <code class='latex inline'>d(t)</code>.<br> b) Describe the transformation that is applied to <code class='latex inline'>f(t)</code> to obtain <code class='latex inline'>d(t)</code>.<br> c) Write the equation that represents the distance of an object from the drop location on each planet.<br> d) Compare the domain and range of each function in part c). What do you notice? </p>
<p>The first three diagrams in a pattern are shown. Each square has a side length of 1 unit. </p><img src="/qimages/22253" /><p>a) Determine the number of squares in each of the next two diagrams, in the pattern. </p><p>b) Make a table comparing the height and area for the five diagrams in your pattern. Use finite differences to determine whether the relation is linear, quadratic, or neither. </p><p>c) Determine an equation for the relationship between the height and the area. </p><p>d) Compare the graph <code class='latex inline'>y=x^2</code> and the graph of the equation from part c). </p>
<img src="/qimages/65677" /><img src="/qimages/65678" />
<p>The triangle with sides given by <code class='latex inline'>x^2 + 1, x^2 -1</code> and <code class='latex inline'>2x</code> will always be right triangle for <code class='latex inline'>x > 1</code>.</p><p>Use the Pythagorean theorem with the expressions for the side lengths to prove that these will always be sides of right triangle for <code class='latex inline'>x > 1</code>.</p>
<p>A rectangle is <code class='latex inline'>7</code> cm longer than it is wide. The diagonal is <code class='latex inline'>13</code> cm. What are the rectangles dimensions?</p>
<p>The maximum viewing distance on a clear day is related to how high you are above the surface of Earth. This relationship can be approximated by the formula <code class='latex inline'>\displaystyle h = \frac{3}{40}d^2 </code>, where <code class='latex inline'>d</code> is the maximum distance, in kilometres, and <code class='latex inline'>h</code> is your height, in metres, above the ground.</p><p>a) How high do you need to be in order to see a distance of 25 km?</p><p>b) How would the formula change if you were standing on a 20 m cliff?</p>
<p>A parabolic arch supporting a bridge</p><p>over a canal is <code class='latex inline'>\displaystyle 10 \mathrm{~m} </code> wide. The height of the arch <code class='latex inline'>\displaystyle 2 \mathrm{~m} </code> from the edge of the canal is <code class='latex inline'>\displaystyle 13 \mathrm{~m} . </code></p><p>a) Determine an equation to represent</p><p>the arch, assuming that the edge of</p><p>the canal is at the origin.</p><p>b) Determine the maximum height</p><p>of the arch, to the nearest tenth of</p><p>a metre.</p>
<p>The population of a town, <code class='latex inline'>\displaystyle P(t) </code>, is modelled by the function <code class='latex inline'>\displaystyle P(t)=6 t^{2}+110 t+3000 </code>, where <code class='latex inline'>\displaystyle t </code> is time in years. Note: <code class='latex inline'>\displaystyle t=0 </code> represents the year <code class='latex inline'>\displaystyle 2000 . </code> a) When will the population reach 6000 ? b) What will the population be in <code class='latex inline'>\displaystyle 2030 ? </code></p>
<p>The population, <code class='latex inline'>P</code>, of a city is modelled by the equation <code class='latex inline'>P = 14t^2 + 820t + 42 000</code>, where tis the time in years.</p><p>When <code class='latex inline'>t = 0</code>, the year is 2008.</p><p>a) Determine the population in 2018.</p><p>b) When was population about 30 000?</p>
<ol> <li>Area rug An area rug has a central <code class='latex inline'>\displaystyle 5 \mathrm{~m} </code> by <code class='latex inline'>\displaystyle 3 \mathrm{~m} </code> rectangle in a mosaic pattern, with a plain border of uniform width around it. The total area of the rug is <code class='latex inline'>\displaystyle 24 \mathrm{~m}^{2} </code>. Find the width of the border.</li> </ol>
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