Now You Try

<p>Express the volume of each solid as a monomial.</p><img src="/qimages/41058" />

<p>The volume, V, in cubic centimetres, of a collection of open-topped boxes can be modelled by
<code class='latex inline'>V(x) = 4x^3 - 200x^2 + 2800x</code>, where x is the height of each box, in centimetres.</p>
<ul>
<li>Fully factor <code class='latex inline'>V(x)</code></li>
</ul>

<p>An artist creates a carving from a rectangular block of soapstone whose volume, <code class='latex inline'>V</code>, in cubic metres, can be modelled by
<code class='latex inline'>V(x) = 6x^3 + 25x2^ + 2x - 8</code>. Determine possible dimensions of the block, in metres, in terms of binomials of <code class='latex inline'>x</code>.</p>

<p>A snowboard manufacturer determines that its profit, P, in thousands
of dollars, can be modelled by the function <code class='latex inline'>P(x) = x +0.00125x^4 -3</code>,
where <code class='latex inline'>x</code> represent the number, in hundreds, of snowboards sold.</p><p>a) What type of function is <code class='latex inline'>P(x)</code>?</p><p>b) Without calculating, determine which </p>
<ul>
<li>i. finite differences are constant for this polynomial function. </li>
<li>ii. What is the value of the constant finite differences? </li>
</ul>
<p>c) Describe the end behaviour of this function, assuming that there are no restrictions on the domain.</p><p>d) State the restrictions on the domain in this situation.</p><p>e) What do the x-intercepts of the graph represent for this situation?</p><p>f) What is the profit from the sale of 3000 snowboards?</p>

<p>A medical researcher establishes that a patient's reaction time, 7, in minutes, to a dose of a particular drug is <code class='latex inline'>r(d) = -0.7d^3 +d^2</code>, where d is the amount of the drug, in millilitres, that is absorbed into the patient's blood.</p>
<ul>
<li>Describe the end behaviour of this function if no restrictions are considered.</li>
</ul>

<p>The volume, V, in cubic centimetres, of a collection of open-topped boxes can be modelled by
<code class='latex inline'>V(x) = 4x^3 - 200x^2 + 2800x</code>, where x is the height of each box, in centimetres.</p>
<ul>
<li>State the value of the constant finite differences for this function.</li>
</ul>

<p>The number, <code class='latex inline'>n</code>, in hundreds, of tent caterpillars infesting a forested area after <code class='latex inline'>t</code> weeks can be modelled by the function <code class='latex inline'>n(t) = -t^4 + 5t^3 + 5t^2 + 6t</code>.</p>
<ul>
<li>What is the population size after 6 weeks?</li>
</ul>

<p>A medical researcher establishes that a patient's reaction time, <code class='latex inline'>r</code>, in minutes, to a dose of a particular drug is <code class='latex inline'>r(d) = -0.7d^3 +d^2</code>, where d is the amount of the drug, in millilitres, that is absorbed into the patient's blood.</p>
<ul>
<li>Without calculating the finite differences, state which finite differences are constant for this function. How do you know? What is the value of the constant differences?</li>
</ul>

<ol>
<li>Aerial flares Red aerial miniflares are used by some boaters in an emergency. The flight of one brand of flare, when fired at an angle of <code class='latex inline'>\displaystyle 70^{\circ} </code> to the horizontal, is modelled by the function <code class='latex inline'>\displaystyle h=-9(t-3)^{2}+83 </code></li>
</ol>
<p>where <code class='latex inline'>\displaystyle h </code> is the height, in metres, and <code class='latex inline'>\displaystyle t </code> is the time, in seconds, since the flare was fired.</p><p>a) What is the maximum height of the flare? b) For how many seconds does the flare burn before it hits the water?</p>

<p>A box has dimension defined by <code class='latex inline'>V =18x^3 -2x + 45x^2 -5</code>.</p><p>a) Determine expressions for the possible dimensions of these boxes.</p><p>b) Determine the dimensions and volume of a box if <code class='latex inline'>x = 2</code> cm</p>

<p>Express the volume of each solid as a monomial.</p><img src="/qimages/41059" />

<p>A storage tank is to be constructed in the shape of a cylinder such that the ratio of the radius, <code class='latex inline'>r</code>, to the height of the tank is 1 :3.</p><p>a) Write a polynomial function to represent</p>
<ul>
<li>i) the surface area of the tank in terms of <code class='latex inline'>r</code></li>
<li>ii) he volume of the tank in terms of <code class='latex inline'>r</code></li>
</ul>
<p>b) Describe the key features of the graph that corresponds to each of the above functions.</p>

<p>The width of a plastic storage box is 1 ft longer than the height. The length is 4 ft longer than the height. The volume is 36 ft". What are the
dimensions of the box?</p>
<ul>
<li>What is the formula for the volume of a rectangular prism?</li>
<li>What variable expressions represent the length, height, and width? </li>
<li>What equation represents the volume of the plastic storage box?</li>
</ul>

<p>The number, <code class='latex inline'>n</code>, in hundreds, of tent caterpillars infesting a forested area after <code class='latex inline'>t</code> weeks can be modelled by the function <code class='latex inline'>n(t) = -t^4 + 5t^3 + 5t^2 + 6t</code>.</p>
<ul>
<li> When is the tent caterpillar population greater than 10 000?</li>
</ul>

<p>Boxes for chocolates are to be constructed from cardboard sheets that measure 36 cm by 20 cm. Each box is formed by folding a sheet along the dotted lines, as shown.</p><img src="/qimages/6954" /><p>a) Express the volume of the box as a function of x.</p><p>b) Determine the possible dimensions of the box if the volume is to be 450 cmâ€œ. Round answers to the nearest tenth of a centimetre.</p><p>c) Write an equation for the family of functions that corresponds to the function
in part a).</p><p>d) Sketch graphs of two members of this family on the same coordinate grid.</p>

<p>A medical researcher establishes that a patient's reaction time, <code class='latex inline'>r</code>, in minutes, to a dose of a particular drug is <code class='latex inline'>r(d) = -0.7d^3 +d^2</code>, where d is the amount of the drug, in millilitres, that is absorbed into the patient's blood.</p><p>State the restrictions for this situation.</p>

<p>Rectangular blocks of limestone are to be cut up and used to build the front entrance of a new hotel. The volume, V, in cubic metres, of each block can be modelled by the function <code class='latex inline'>V(x) = 2x^3 + 7x^2 + 2x - 3</code>.</p><p>a) Determine the dimensions of the blocks in terms of <code class='latex inline'>x</code>.</p><p>b) What are the possible dimensions of the blocks when <code class='latex inline'>x =1</code>?</p>

<p>Express the volume of each solid as a monomial.</p><img src="/qimages/41060" />

<p>A rectangular box is <code class='latex inline'>2x + 3</code> units long, <code class='latex inline'>2x â€” 3</code> units wide, and 3x units high. What is its volume, expressed as a polynomial?</p>

<p>A ball with a 3 in. radius has volume <code class='latex inline'>\displaystyle V_{1} . </code> A second ball has a 9 in. radius and volume <code class='latex inline'>\displaystyle V_{2} </code>. Which equation represents the volume of the second ball in terms of the first?</p><p>(A) <code class='latex inline'>\displaystyle V_{2}=3 V_{1} </code></p><p>(B) <code class='latex inline'>\displaystyle V_{2}=27 V_{1} </code></p><p>(C) <code class='latex inline'>\displaystyle V_{2}=V_{1}^{2} </code></p><p>(D) <code class='latex inline'>\displaystyle V_{2}=9 V_{1}^{2} </code></p>

<p>A medical researcher establishes that a patient's reaction time, <code class='latex inline'>r</code>, in minutes, to a dose of a particular drug is <code class='latex inline'>r(d) = -0.7d^3 +d^2</code>, where d is the amount of the drug, in millilitres, that is absorbed into the patient's blood.</p>
<ul>
<li>What type of function is <code class='latex inline'>r(d)</code>?</li>
</ul>

<p>A section of a water tube ride at an amusement park can be modelled by the function</p><p><code class='latex inline'>h(t) = -0.002t^4 + 0.104t^3 -1.69t^2 + 8.5t + 9</code>,</p><p>where <code class='latex inline'>t</code> is the time, in seconds, and <code class='latex inline'>h</code> is the height, in metres, above the ground. When will the riders be more than <code class='latex inline'>15</code> m above the ground?</p><p><strong>You may use a graphing device for this question</strong></p>

<p>An open-topped box is made from a rectangular piece of cardboard, with dimensions of <code class='latex inline'>24</code> cm by <code class='latex inline'>30</code> cm, by cutting congruent squares from each corner and folding up the sides. Determine the dimensions of the squares to be cut to create a box with a volume of <code class='latex inline'>1040 cm^3</code></p>

<p>The eBox marketing team has determine that the number, c, in thousands, of customers who purchase the company's products on-line from the eBox web site t years after 2003 can be modelled by the function <code class='latex inline'>c(t) = 0.1t^3 - 2t + 8.</code></p><p><strong>(a)</strong> When will there be fewer than 8000 on-line customers?</p><p><strong>(b)</strong> When will the number of on-line customers exceed 10 000?</p><p>You may use a graphing device for this question.</p>

<p>The volume in cubic feet of a CD holder can be expressed as <code class='latex inline'>V(x) = -x^3 â€” x^2 + 6x</code>, or, when factored, as the product of its three dimensions. The depth is expressed as <code class='latex inline'>2 â€” x</code>. Assume that the height is greater than the width.</p><p>a. Factor the polynomial to find linear expressions for the height and the width. </p><p>b. Graph the function. Find the x-intercepts. What do they represent?</p><p>c. What is a realistic domain for the function?</p><p>d. What is the maximum volume of the CD holder?</p>

<p>The product of three consecutive integers is 210. What are the numbers?</p>

<img src="/qimages/63577" /><p>Write, expand,</p><p>and simplify an expression that</p><p>represents the surface area of the</p><p>rectangular prism.</p>

<p>Measurement The dimensions of a rectangular prism are represented by binomials, as shown.</p><p>a) Write, expand, and simplify an expression that represents the surface area of this prism.</p><p>b) If <code class='latex inline'>\displaystyle x </code> represents <code class='latex inline'>\displaystyle 5 \mathrm{~cm} </code>, what is the surface area, in square centimetres?</p>

<p>Think About a Plan A storage company needs to design a new storage box that has twice the volume of its largest box. Its largest box is <code class='latex inline'>\displaystyle 5 \mathrm{ft} </code> long, <code class='latex inline'>\displaystyle 4 \mathrm{ft} </code> wide, and <code class='latex inline'>\displaystyle 3 \mathrm{ft} </code> high. The new box must be formed by increasing each dimension by the same amount. Find the increase in each dimension.</p>
<ul>
<li><p>How can you write the dimensions of the new storage box as polynomial expressions?</p></li>
<li><p>How can you use the volume of the current largest box to find the volume of the new box?</p></li>
</ul>

<p>Which expression is the factored form of <code class='latex inline'>\displaystyle x^{3}+2 x^{2}-5 x-6 ? </code></p><p>F) <code class='latex inline'>\displaystyle (x+1)(x+1)(x-6) </code></p><p>(H) <code class='latex inline'>\displaystyle (x+2)(2 x-5)(x-6) </code></p><p>(G) <code class='latex inline'>\displaystyle (x+3)(x+1)(x-2) </code></p><p>1) <code class='latex inline'>\displaystyle (x-3)(x-1)(x+2) </code></p><p>A ball with a 3 in. radius has volume <code class='latex inline'>\displaystyle V_{1} . </code> A second ball has a 9 in. radius and volume <code class='latex inline'>\displaystyle V_{2} </code>. Which equation represents the volume of the second ball in terms of the first?</p><p>(A) <code class='latex inline'>\displaystyle V_{2}=3 V_{1} </code></p><p>(B) <code class='latex inline'>\displaystyle V_{2}=27 V_{1} </code></p><p>(C) <code class='latex inline'>\displaystyle V_{2}=V_{1}^{2} </code></p><p>(D) <code class='latex inline'>\displaystyle V_{2}=9 V_{1}^{2} </code></p><p>What is the polynomial function, in factored form, whose zeros are <code class='latex inline'>\displaystyle -2,5 </code>, and 6 , and whose leading coefficient is <code class='latex inline'>\displaystyle -2 ? </code> Graph this function and find any relative</p>

<p>The length of an edge of a cube is represented by the expression <code class='latex inline'>\displaystyle 2 x-y . </code> Write, expand, and simplify an expression that represents the surface area of the cube.</p>

<p>A demographer develops a model for the population, P, of a small town years from today such that <code class='latex inline'>P(n) = -0.15n^5 + 3n^4 + 5560.</code></p><p>Will the model be valid after 20 years? Explain.</p>

<p>A open top box is to be constructed from a piece of cardboard by cutting congruent squares from the corners and then folding up the sides.
The dimensions of the cardboard are shown.</p><img src="/qimages/6955" /><p>a) Express the volume of the box as a function of <code class='latex inline'>x</code>.</p><p>b) Write an equation that represents the box with volume</p>
<ul>
<li>i) twice the volume fo the box represented by the function in part a)</li>
<li>ii) half the volume of the box represented by the function in part a)</li>
</ul>
<p>c) Use your function from part a) to determine the values of <code class='latex inline'>x</code> that will result in boxes with a volume greater than 2016 <code class='latex inline'>cm^3</code>.</p>

<p>Between 1985 through 1995, the number of home computers, in thousands, sold in Canada is estimated by <code class='latex inline'>C(t) = 0.92(t^3 + 8t^2+40t + 400)</code>, </p><p>where t is in years and <code class='latex inline'>t = 0</code> for <code class='latex inline'>1985</code>.</p><p>a) Explain why you can use this model to predict the number of home computes sold in 1993, but not to predict sales in 2005. </p><p>b) Explain how to find when the number of home computers sales in Canada reached 1.5 million, using this model.</p><p>c) In what year did home computer sales reach 1.5 million?</p>

<p>A carpenter hollowed out the interior of a block of wood as shown at the right.</p><p>a. Express the volume of the original block and the volume of the
wood removed as polynomials in factored form.</p><p>b. What polynomial represents the volume of the wood remaining?</p><img src="/qimages/23912" />