Now You Try

<p>The main character in a video game, Tammy, must swing on a vine to
cross a river. If she grabs the vine at a point that is too low and swings
within 80 cm of the surface of the river, a crocodile will come out of the
river and catch her. From where she is standing on the riverbank,
Tammy can reach a point on the vine where her height above the river,
<code class='latex inline'>h</code>, is modelled by the relation <code class='latex inline'>h=12x^2-76.8x+198</code>, where <code class='latex inline'>x</code> is the horizontal distance of her swing from her starting point. Should
Tammy jump? Justify your answer.</p>

<p>The population of a Canadian city is modelled by <code class='latex inline'>P(t) = 12t^2 + 800t + 40 000</code>, where t is the time in year. When t = 0, the year is 2007.</p>
<ul>
<li>According to the model, what will the population be in 2020?</li>
</ul>

<p>Create a quadratic model for the height of a toy rocket launched upward at <code class='latex inline'>45</code> m/s from a <code class='latex inline'>2</code>-m platform.</p>

<p>Chelsea has just started her own dog-grooming business. On the first day, she groomed four dogs for a profit of <code class='latex inline'>\$26.80</code>. On the second day, she groomed 15 dogs for a profit of <code class='latex inline'>\$416.20</code>. She thinks that she will maximize her profit if she grooms 11 dogs per day. Assuming that her profit can be modelled by a quadratic relation, calculate her maximum profit.</p>

<p>A bus company has <code class='latex inline'>4000</code> passengers daily, each paying a fare of $<code class='latex inline'>2</code>. For each $<code class='latex inline'>0.15</code> increase, the company estimates that it will lose <code class='latex inline'>40</code> passengers per day. If the company needs to take in <code class='latex inline'>\$10 450</code> per day to stay in business, what fare should be charged?</p>

<p>A rectangular field measures 15 m by 20 m. A rectangular area is to be fenced in by reducing each dimension by the same 1 amount. The fenced-in area will be <code class='latex inline'>\displaystyle{\frac{1}{2}}</code> the original area. What will the dimensions of the fenced-in area be? Include a diagram in your solution.</p>

<p>Jackie mows a strip of uniform width around her <code class='latex inline'>25</code> m by <code class='latex inline'>15</code> m rectangular lawn and leaves a patch of lawn that is <code class='latex inline'>60\%</code> of the original area. What is the <em>width</em> of the strip?</p>

<p>What is the perimeter of a right triangle with legs 6 cm and 3 cm? Leave your answer in simplest radical form.</p>

<p>A rectangular construction site is enclosed on three sides using <code class='latex inline'>1200</code> m of fencing. The remaining side is formed by an existing wall. What dimensions enclose <code class='latex inline'>180 000</code> m<code class='latex inline'>^2</code> of land?</p><img src="/qimages/1123" />

<p>An open-topped box is to be made from a rectangular piece of tin measuring <code class='latex inline'>50</code> cm by <code class='latex inline'>40</code> cm by cutting squares of equal size from each corner. The base area is to be 875 <code class='latex inline'>cm^2</code></p><p><strong>a)</strong> Draw a diagram representing the information.</p><p><strong>b)</strong> What is the side length of the squares being removed?</p><p><strong>c)</strong> What is the volume of the box?</p>

<p>The revenue function for a production by a theatre group is <code class='latex inline'>R(t) = -50t^2 + 300t</code>, where t is the ticket price in dollars. The cost function for the production is <code class='latex inline'>C(t) = 600 - 50t</code>. Determine the ticket price that will allow the production to break even.</p>

<p>The flight of a ball hit from a tee that is 0.6m tall can be modelled by the function <code class='latex inline'>h(t) = -4.9t^2 + 6t + 0.6</code> where <code class='latex inline'>h(t)</code> is the height in metres at time <code class='latex inline'>t</code> seconds. How long will it take for the ball to hit the ground? </p>

<p>Joan kicked a soccer ball. The height of the ball, <code class='latex inline'>h</code>, in meters, can be modelled by <code class='latex inline'>h=-1.2x^2+6x</code>, where <code class='latex inline'>x</code> is the horizontal distance,
in meters, from where she kicked the ball.</p>
<ul>
<li>What was the initial height of the ball when she kicked it?
How do you know?</li>
</ul>

<p>In the TV show Junkyard Wars, a trebuchet was used to catapult a pumpkin from a height of <code class='latex inline'>4</code> m for a total horizontal distance of <code class='latex inline'>24</code> m. It reached a maximum height of <code class='latex inline'>14</code> m. At what horizontal distances was the height of the pumpkin <code class='latex inline'>10</code> m, to the nearest metre?</p>

<p>A photograph measures <code class='latex inline'>21</code> cm by <code class='latex inline'>15</code> cm. A strip of constant width is to be cut from each side of the photo. so the area is reduced to 216 <code class='latex inline'>cm^2</code>. Find the width of the cut. Include a diagram in your solution.</p>

<p>A firework is launched upward at an initial velocity of <code class='latex inline'>49</code> m/s, from a height of <code class='latex inline'>1.5</code> m above the ground. The height of the firework, in metres, after <code class='latex inline'>t</code> seconds, is modelled by the equation <code class='latex inline'>h=-4.9t^2+49t+1.5</code></p><p>a) What is the maximum height of the firework above the ground?</p>

<p>Carly has just opened her own nail salon. Based on experience, she
knows that her daily profit, <code class='latex inline'>P</code>, in dollars, can be modelled by the relation <code class='latex inline'>P=-15x^2+240x-640</code>, where <code class='latex inline'>x</code> is the number of clients per day.</p><p>How many clients should she book each day to maximize her profit?</p>

<p>Joan kicked a soccer ball. The height of the ball, <code class='latex inline'>h</code>, in meters, can be modelled by <code class='latex inline'>h=-1.2x^2+6x</code>, where <code class='latex inline'>x</code> is the horizontal distance,
in meters, from where she kicked the ball.</p><p><strong>i.</strong> Complete the square to write the relation in vertex form.</p><p><strong>ii.</strong> State the vertex of the relation,</p><p><strong>iii.</strong> What does each coordinate of the vertex represent in this situation?</p><p><strong>iv.</strong> How far did Joan kick the ball?</p>

<p>A rectangle has an area of 330 <code class='latex inline'>m^2</code>. One side is <code class='latex inline'>7</code> m longer than the other. What are the dimensions of the rectangle? </p>

<p>Determine the number of points of intersection of each pair of parabolas. Justify your answer.</p>
<ul>
<li><code class='latex inline'>\displaystyle
y = 3x^2 - 12x + 16
</code></li>
<li><code class='latex inline'>\displaystyle
y = -2x^2 -4x + 3
</code></li>
</ul>

<p>A rectangle has perimeter <code class='latex inline'>23</code> <code class='latex inline'>cm</code>. Its area is <code class='latex inline'>33</code> <code class='latex inline'>cm^2</code>. Determine the dimensions of the rectangle. Include a diagram in your solution.</p>

<p>The cost, <code class='latex inline'>C</code>, in dollars, to hire landscapers to weed and seed a local
park can be modelled by <code class='latex inline'>C=6x^2-60x+900</code>, where <code class='latex inline'>x</code> is the
number of landscapers hired to do the work. </p><p>How many landscapers should be hired to minimize the cost?</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 swing above the ground
during one swing, <code class='latex inline'>t</code> seconds after the swing begins to move forward</li>
</ul>

<p>A cylindrical can with height <code class='latex inline'>12</code> cm has capacity <code class='latex inline'>600</code> mL. What is its radius. to the nearest millimetre? [Remember that <code class='latex inline'>1 mL = 1 cm^3</code>.]</p><img src="/qimages/1121" />

<p>The height, <code class='latex inline'>h(t)</code>, of a baseball, in metres, at time t seconds after it is tossed out of a window is modelled by the function <code class='latex inline'>h(t) = -5t^2+20t + 15</code>. A boy shoots at the baseball with a paintball gun. The trajectory of the paintball is given by the function <code class='latex inline'>g(t) = 3t + 3</code>. Will the paintball hit the baseball? If so, when? At what height will the baseball be?</p>

<p>How long would the rocket take to fall to Earth, rounded to the nearest hundredth of a second?</p>

<p>The area of a triangle is <code class='latex inline'>20</code> cm<code class='latex inline'>^2</code>, and the altitude is 4 cm greater than the base. Find the length of the base. to the nearest millimetre.</p><img src="/qimages/1122" />

<p>A ladder is <code class='latex inline'>6</code> m long. If the height of the top of the ladder must be no greater than <code class='latex inline'>10</code> times the distance from the base to the wall, how high up a wall can the top of the ladder be placed? Include a diagram in your solution. Round to the nearest millimetre.</p>

<p>A rotating liquid surface takes on the shape of a parabolic mirror. The diameter of the mirror is <code class='latex inline'>6</code> m. The vertex is <code class='latex inline'>23</code> cm below the edges. Find an equation to model the parabolic cross section of the mirror. </p>

<p>Determine the number of points of intersection of each pair of parabolas. Justify your answer.</p>
<ul>
<li><code class='latex inline'>\displaystyle
y = x^2 + 2x + 7
</code></li>
<li><code class='latex inline'>\displaystyle
y = x^2 -4x -1
</code></li>
</ul>

<p>A golf ball is hit, and it lands at a point on the same horizontal plane <code class='latex inline'>53</code> m away. The path of the ball took it just over a 9m tall tree that was <code class='latex inline'>8</code>m in front of the golfer.</p><p><strong>(a)</strong> Assume the ball is hit from the origin of a coordinate plane. Find a quartic function that describes the path of the ball.</p><p><strong>(b)</strong> What is the maximum height of the ball?</p><p><strong>(c)</strong> Is possible to move the origin in this situation on and develop another quadratic function to describe the path? If so, find a second quartic function.</p>

<p>A rectangular field with an area of <code class='latex inline'>8000 m^2</code> is enclosed by 400 m of fencing. Determine the dimensions of the filed to the nearest tenth of a metre.</p>

<p>Kayli wants to build a parabolic bridge over a stream in her backyard as
shown at the left. The bridge must span a width of 200 cm. It must be at
least 51 cm high where it is 30 cm from the bank on each side. How
high will her bridge be?</p><img src="/qimages/863" />

<p>A rectangular garden measures <code class='latex inline'>15</code> m by <code class='latex inline'>24</code> m. A larger garden is to be made by increasing each side length by the same amount. The resulting area is to be <code class='latex inline'>1.5</code> times the original area. Find the dimensions of the new garden, to the nearest tenth of a metre. Include a diagram in your solution.</p>

<p>The length of one leg of a right triangle is <code class='latex inline'>7</code> cm more than that of the other leg. The length of the hypotenuse is <code class='latex inline'>3</code> cm more than double that of the shorter leg. Find the lengths of each of the three sides.</p><img src="/qimages/1120" />

<p>The length of a rectangular field is <code class='latex inline'>2</code> m greater than three times its width. The area of the field is 1496 <code class='latex inline'>m^2</code>. What are the dimensions of the field?</p>

<p>Determine the break-even points of the profit function <code class='latex inline'>P(x) = -2x^2 + 7x + 8</code>, where <code class='latex inline'>x</code> is the number of dirt bikes reduced, in thousands.</p>

<p>A engineer is designing a parabolic arch. The arch must be 15 m high, and 6 m wide at a height of 8 m. </p><p>Determine a quadratic function that satisfies these conditions.</p>

<p>The population of a Canadian city is modelled by <code class='latex inline'>P(t) = 12t^2 + 800t + 40 000</code>, where t is the time in year. When t = 0, the year is 2007.</p>
<ul>
<li>In what year is the population predicted to be 300 000?</li>
</ul>

<p>An integer is two more than another integer. Twice the larger integer is one more than the square of the smaller integer. Find the two integers.</p>

<p>A right triangle has a height of <code class='latex inline'>8</code> cm more than twice the length of the base. If the area of the triangle is <code class='latex inline'>96</code> <code class='latex inline'>cm^2</code>, what are the dimensions of the triangle? </p>

<p>The perimeter of a right triangle is 60 cm. The length of the hypotenuse is 6 cm more than twice the length of one of the other sides. Find the lengths of all three sides.</p>

<p>A science experiment involves launching a small rocket. The following measurements are taken:</p><p>Initial height: <code class='latex inline'>0.61</code> m</p><p>Initial vertical velocity: <code class='latex inline'>36.85</code> m/s</p><p><strong>a)</strong> Create a quadratic model for the height, in metres, of the rocket after a given number of seconds.</p><p><strong>b)</strong> Verify the following results of the experiment:</p><p>Total time in the air: <code class='latex inline'>7.54</code> s</p><p>Maximum height: <code class='latex inline'>69.89</code> m</p><p><strong>c)</strong> Sketch a graph of this relation and label the key information as in Example 1 of this section.</p>

<p>A rectangular picture frame measures <code class='latex inline'>20</code> cm by <code class='latex inline'>30</code> cm. A new frame is to be made by increasing each side length by the same amount. </p><p>The resulting enclosed area is to be 1064 <code class='latex inline'>cm^2</code>. Find the dimensions of the new picture frame. Include a diagram in your solution.</p>

<p>Determine the number of points of intersection of each pair of parabolas. Justify your answer.</p>
<ul>
<li><code class='latex inline'>\displaystyle
y = x^2 - 6x + 10
</code></li>
<li><code class='latex inline'>\displaystyle
y = 5x^2 -30x + 46
</code></li>
</ul>

<p>A firework is launched upward at an initial velocity of 49 m/s, from a height of 1.5 m above the ground. The height of the firework, in metres, after <code class='latex inline'>t</code> seconds, is modelled by the equation <code class='latex inline'>h=-4.9t^2+49t+1.5</code></p><p>Over what time interval is the height of the firework greater than <code class='latex inline'>100</code> m above the ground? Round to the nearest hundredth of a second.</p>

<p>Bob wants to cut a Wire that is 60 cm long into two pieces. Then he wants to make each piece into a square. Determine how the wire should be cut so that the total area of the two squares is as small as possible.</p>

<p>The length of a rectangle is <code class='latex inline'>16</code> m greater than its width. The area is <code class='latex inline'>35</code><code class='latex inline'>m^2</code>. Find the dimensions of the rectangle, to the nearest hundredth of a metre.</p><img src="/qimages/1119" />

<p>A photograph measures <code class='latex inline'>20</code> cm by <code class='latex inline'>16</code> cm. A strip of constant width is to be cut off the top and one side of the photo, so the area is reduced to <code class='latex inline'>60\%</code> of the area of the original photo. Find the width of the cut. Include a diagram in your solution.</p>

<p>A engineer is designing a parabolic arch. The arch must be 15 m high, and 6 m wide at a height of 8 m. </p><p>What its the width of the arch at its base?</p>