chap3-rev-problems.xml

<?xml version="1.0"?>


<!-- This file was originally part of the book     -->
<!--   Modeling, Functions, and Graphs   -->
<!--           4th                                     -->
<!-- Copyright (C) Katherine Yoshiwara      -->

<exercises xml:id="chap3-rev-problems"  xmlns:xi="http://www.w3.org/2001/XInclude">
    <title>Review Problems</title>


<exercise number="1">
    <statement><p>
        The distance s a pebble falls through a thick liquid varies directly with the square of the length of time <m>t</m> it falls.
        <ol label="a">
            <li><p>If the pebble falls <m>28</m> centimeters in <m>4</m> seconds, express the distance it will fall as a function of time.</p></li>
            <li><p>Find the distance the pebble will fall in <m>6</m> seconds.</p></li>
        </ol>
    </p></statement>
    <answer><p><ol label="a">
        <li><p><m>d = 1.75t^2</m></p></li>
        <li><p><m>63</m> cm</p></li>
    </ol></p></answer>
</exercise>

<exercise number="2">
    <statement><p>
        The volume, <m>V</m>, of a gas varies directly with the temperature, <m>T</m>, and inversely with the pressure, <m>P</m>, of the gas.
        <ol label="a">
            <li><p>If <m>V = 40</m> when <m>T = 300</m> and <m>P = 30</m>, express the volume of the gas as a function of the temperature and pressure of the gas.</p></li>
            <li><p>Find the volume when <m>T = 320</m> and <m>P = 40</m>.</p></li>
        </ol>
    </p></statement>
</exercise>

<exercise number="3">
    <statement><p>The demand for bottled water is inversely proportional to the price per bottle. If Droplets can sell <m>600</m> bottles at <m>\$8</m> each, how many bottles can the company sell at <m>\$10</m> each?</p></statement>
<answer><p><m>480</m> bottles </p></answer>
</exercise>

<exercise number="4">
    <statement><p>The intensity of illumination from a light source varies inversely with the square of the distance from the source. If a reading lamp has an intensity of <m>100</m> lumens at a distance of <m>3</m> feet, what is its intensity <m>8</m> feet away?</p></statement>
</exercise>

<exercise number="5">
    <statement><p>
        A person's weight, <m>w</m>, varies inversely with the square of his or her distance, <m>r</m>, from the center of the Earth.
        <ol label="a">
            <li><p>Express <m>w</m> as a function of <m>r</m>. Let <m>k</m> stand for the constant of variation.</p></li>
            <li><p>Make a rough graph of your function.</p></li>
            <li><p>How far from the center of the Earth must Neil be in order to weigh one-third of his weight on the surface? The radius of the Earth is about <m>3960</m> miles.</p></li>
        </ol>
    </p></statement>
    <answer><p><ol label="a">
        <li><p><m>w = \dfrac{k}{r^2}</m></p></li>
        <li><p><image source="images/fig-ans-chap3-rev-5"  width="35%"><description>inverse square in first quadrant</description></image> </p></li>
        <li><p><m>3960\sqrt{3}\approx 6860</m> miles</p></li>
    </ol></p></answer>
</exercise>

<exercise number="6">
   <statement><p>
        The period, <m>T</m>, of a pendulum varies directly with the square root of its length, <m>L</m>.
        <ol label="a">
            <li><p>Express <m>T</m> as a function of <m>L</m>. Let <m>k</m> stand for the constant of variation.</p></li>
            <li><p>Make a rough graph of your function.</p></li>
            <li><p>If a certain pendulum is replaced by a new one four-fifths as long as the old one, what happens to the period?</p></li>
        </ol>
    </p></statement>
</exercise>


<exercisegroup cols="4"><introduction><p>
    For each table, <m>y</m> varies directly or inversely with a power of <m>x</m>. Find the power of <m>x</m> and the constant of variation, <m>k</m>. Write a formula for each function of the form <m>y = kx^n</m> or <m>y = \dfrac{k}{x^n}</m>.
</p></introduction>

<exercise number="7"><statement><p>
    <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>y</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>4.8</m></cell>
        </row>
        <row>
            <cell><m>5</m></cell>
            <cell><m>30.0</m></cell>
        </row>
        <row>
            <cell><m>8</m></cell>
            <cell><m>76.8</m></cell>
        </row>
        <row>
            <cell><m>11</m></cell>
            <cell><m>145.2</m></cell>
        </row>
</tabular></p></statement>
<answer><p><m>y = 1.2x^2</m></p></answer>
</exercise>

<exercise number="8"><statement><p>
    <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>y</m></cell>
        </row>
        <row>
            <cell><m>1.4</m></cell>
            <cell><m>75.6</m></cell>
        </row>
        <row>
            <cell><m>2.3</m></cell>
            <cell><m>124.2</m></cell>
        </row>
        <row>
            <cell><m>5.9</m></cell>
            <cell><m>318.6</m></cell>
        </row>
        <row>
            <cell><m>8.3</m></cell>
            <cell><m>448.2</m></cell>
        </row>
</tabular></p></statement>
</exercise>

<exercise number="9"><statement><p>
    <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>y</m></cell>
        </row>
        <row>
            <cell><m>0.5</m></cell>
            <cell><m>40.0</m></cell>
        </row>
        <row>
            <cell><m>2.0</m></cell>
            <cell><m>10.0</m></cell>
        </row>
        <row>
            <cell><m>4.0</m></cell>
            <cell><m>5.0</m></cell>
        </row>
        <row>
            <cell><m>8.0</m></cell>
            <cell><m>2.5</m></cell>
        </row>
</tabular></p></statement>
<answer><p><m>y =\dfrac{20}{x} </m></p></answer>
</exercise>

<exercise number="10"><statement><p>
    <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell><m>x</m></cell>
            <cell><m>y</m></cell>
        </row>
        <row>
            <cell><m>1.5</m></cell>
            <cell><m>320.0</m></cell>
        </row>
        <row>
            <cell><m>2.5</m></cell>
            <cell><m>115.2</m></cell>
        </row>
        <row>
            <cell><m>4.0</m></cell>
            <cell><m>45.0</m></cell>
        </row>
        <row>
            <cell><m>6.0</m></cell>
            <cell><m>20.0</m></cell>
        </row>
</tabular></p></statement>
</exercise>
</exercisegroup>


<exercisegroup ><introduction><p>
    Write without negative exponents and simplify.
</p></introduction>

<exercise number="11"><statement><p><ol label="a" cols="2">
    <li><p><m>(-3)^{-4} </m></p></li>
    <li><p><m>4^{-3}</m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\dfrac{1}{81} </m></p></li>
    <li><p><m>\dfrac{1}{64} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="12"><statement><p><ol label="a" cols="2">
    <li><p><m>\left(\dfrac{1}{3}\right)^{-2} </m></p></li>
    <li><p><m>\dfrac{3}{5^{-2}} </m></p></li>
</ol></p></statement>
</exercise>

<exercise number="13"><statement><p><ol label="a" cols="2">
    <li><p><m>(3m)^{-5} </m></p></li>
    <li><p><m>-7y^{-8}</m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\dfrac{1}{243m^5} </m></p></li>
    <li><p><m>\dfrac{-7}{y^8} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="14"><statement><p><ol label="a" cols="2">
    <li><p><m>a^{-1}+ a^{-2} </m></p></li>
    <li><p><m>\dfrac{3q^{-9}}{r^{-2}} </m></p></li>
</ol></p></statement>
</exercise>

<exercise number="15"><statement><p><ol label="a" cols="2">
    <li><p><m>6c^{-7}\cdot (3)^{-1} c^4 </m></p></li>
    <li><p><m>\dfrac{11z^{-7}}{3^{-2} z^{-5}}</m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\dfrac{2}{c^3}  </m></p></li>
    <li><p><m>\dfrac{99}{z^2} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="16"><statement><p><ol label="a" cols="2">
    <li><p><m>\left(2d^{-2}k^3 \right)^{-4} </m></p></li>
    <li><p><m>\dfrac{2w^3(w^{-2})^{-3}}{5w^{-5}} </m></p></li>
</ol></p></statement>
</exercise>
</exercisegroup>


<exercisegroup><introduction><p>Write each power in radical form.</p></introduction>

<exercise number="17"><statement><p><ol label="a" cols="2">
    <li><p><m>25m^{1/2} </m></p></li>
    <li><p><m>8n^{-1/3} </m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>25\sqrt{m} </m></p></li>
    <li><p><m>\dfrac{8}{\sqrt[3]{n}} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="18"><statement><p><ol label="a" cols="2">
    <li><p><m>(13d)^{2/3} </m></p></li>
    <li><p><m>6x^{2/5}y^{3/5} </m></p></li>
</ol></p></statement>
</exercise>

<exercise number="19"><statement><p><ol label="a" cols="2">
    <li><p><m>(3q)^{-3/4} </m></p></li>
    <li><p><m>7(uv)^{3/2} </m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>\dfrac{1}{\sqrt[4]{27q^3}} </m></p></li>
    <li><p><m>7\sqrt{u^3v^3} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="20"><statement><p><ol label="a" cols="2">
    <li><p><m>(a^2+b^2)^{0.5} </m></p></li>
    <li><p><m>(16-x^2)^{0.25} </m></p></li>
</ol></p></statement>
</exercise>
</exercisegroup>


<exercisegroup><introduction><p>Write each radical as a power with a fractional exponent.</p></introduction>

<exercise number="21"><statement><p><ol label="a" cols="2">
    <li><p><m>2\sqrt[3]{x^2} </m></p></li>
    <li><p><m>\dfrac{1}{4}\sqrt[4]{x} </m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>2x^{2/3} </m></p></li>
    <li><p><m>\dfrac{1}{4}x^{1/4} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="22"><statement><p><ol label="a" cols="2">
    <li><p><m>z^2\sqrt{z} </m></p></li>
    <li><p><m>z\sqrt[3]{z} </m></p></li>
</ol></p></statement>
</exercise>

<exercise number="23"><statement><p><ol label="a" cols="2">
    <li><p><m>\dfrac{6}{\sqrt[4]{b^3}} </m></p></li>
    <li><p><m>\dfrac{-1}{3\sqrt[3]{b}} </m></p></li>
</ol></p></statement>
<answer><p><ol label="a" cols="2">
    <li><p><m>6b^{-3/4} </m></p></li>
    <li><p><m>\dfrac{-1}{3}b^{-1/3} </m></p></li>
</ol></p></answer>
</exercise>

<exercise number="24"><statement><p><ol label="a" cols="2">
    <li><p><m>\dfrac{-4}{(\sqrt[4]{b^3})} </m></p></li>
    <li><p><m>\dfrac{2}{(\sqrt{a})^3} </m></p></li>
</ol></p></statement>
</exercise>
</exercisegroup>


<exercisegroup><introduction><p>Sketch graphs by hand for each function on the domain <m>(0,\infty)</m>.</p></introduction>

<exercise number="25"><statement><p>
    <m>y</m> varies directly with <m>x^2</m>. The constant of variation is <m>k = 0.25</m>.
</p></statement>
<answer><p><image source="images/fig-ans-chap3-rev-25"  width="35%"><description>inverse square</description></image> </p></answer>
</exercise>

<exercise number="26"><statement><p>
    <m>y</m> varies directly with <m>x</m>. The constant of variation is <m>k = 1.5</m>.
</p></statement></exercise>

<exercise number="27"><statement><p>
    <m>y</m> varies inversely with <m>x</m>. The constant of variation is <m>k = 2</m>.
</p></statement>
<answer><p><image source="images/fig-ans-chap3-rev-27"  width="35%"><description>inverse square</description></image> </p></answer>
</exercise>

<exercise number="28"><statement><p>
    <m>y</m> varies inversely with <m>x^2</m>. The constant of variation is <m>k = 4</m>.
</p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>
    Write each function in the form <m>y = kx^p</m>.
</p></introduction>

<exercise number="29"><statement><p><m>f(x)=\dfrac{2}{3x^4} </m></p></statement>
<answer><p><m>f(x)=\dfrac{2}{3}x^{-4} </m></p></answer>
</exercise>

<exercise number="30"><statement><p><m>g(x)=\dfrac{8x^7}{29} </m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>For Problems 31–34,
    <ol label="(a)">
        <li><p>Evaluate each function for the given values.</p></li>
        <li><p>Graph the function.</p></li>
    </ol>
</p></introduction>

<exercise number="31"><statement><p>
    <m>Q(x)=4x^{5/2} </m></p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell halign="left"><m>x</m></cell>
            <cell><m>16</m></cell>
            <cell><m>\dfrac{1}{4} </m></cell>
            <cell><m>3</m></cell>
            <cell><m>100</m></cell>
        </row>
        <row>
            <cell halign="left"><m>Q(x)</m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
<answer><p><ol label="(a)">
    <li><p>
            <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
                <row>
                    <cell halign="left"><m>x</m></cell>
                    <cell><m>16</m></cell>
                    <cell><m>\dfrac{1}{4} </m></cell>
                    <cell><m>3</m></cell>
                    <cell><m>100</m></cell>
                </row>
                <row>
                    <cell halign="left"><m>Q(x)</m></cell>
                    <cell><m>4096 </m></cell>
                    <cell><m>\dfrac{1}{8} </m></cell>
                    <cell><m>4\sqrt{3^5}\approx 62.35 </m></cell>
                    <cell><m>400,000</m></cell>
                </row>
            </tabular>
    </p></li>
    <li><p><image source="images/fig-ans-chap3-rev-31"  width="35%"><description>increasing concave up</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="32"><statement><p>
    <m>T(w)=-3w^{2/3} </m></p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell halign="left"><m>w</m></cell>
            <cell><m>27</m></cell>
            <cell><m>\dfrac{1}{8} </m></cell>
            <cell><m>20</m></cell>
            <cell><m>1000</m></cell>
        </row>
        <row>
            <cell halign="left"><m>T(w) </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
</exercise>

<exercise number="33"><statement><p>
    <m>f(x)=x^{0.3} </m></p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell halign="left"><m>x</m></cell>
            <cell><m>0</m></cell>
            <cell><m>1 </m></cell>
            <cell><m>5</m></cell>
            <cell><m>10</m></cell>
            <cell><m>20</m></cell>
            <cell><m>50</m></cell>
            <cell><m>70</m></cell>
            <cell><m>100</m></cell>
        </row>
        <row>
            <cell halign="left"><m>f(x)</m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
<answer><p><ol label="(a)">
    <li><p>
            <tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
                <row>
                    <cell halign="left"><m>x</m></cell>
                    <cell><m>0</m></cell>
                    <cell><m>1 </m></cell>
                    <cell><m>5</m></cell>
                    <cell><m>10</m></cell>
                    <cell><m>20</m></cell>
                    <cell><m>50</m></cell>
                    <cell><m>70</m></cell>
                    <cell><m>100</m></cell>
                </row>
                <row>
                    <cell halign="left"><m>f(x)</m></cell>
                    <cell><m>0 </m></cell>
                    <cell><m>1 </m></cell>
                    <cell><m>1.62</m></cell>
                    <cell><m>2.00 </m></cell>
                    <cell><m>2.46 </m></cell>
                    <cell><m>3.23</m></cell>
                    <cell><m>3.58</m></cell>
                    <cell><m>3.98</m></cell>
                </row>
            </tabular>
    </p></li>
    <li><p><image source="images/fig-ans-chap3-rev-33"  width="35%"><description>increasing concave up</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="34"><statement><p>
    <m>g(x)=-x^{-0.7} </m></p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center">
        <row>
            <cell halign="left"><m>x</m></cell>
            <cell><m>0.1</m></cell>
            <cell><m>0.2 </m></cell>
            <cell><m>0.5</m></cell>
            <cell><m>1</m></cell>
            <cell><m>5</m></cell>
            <cell><m>8</m></cell>
            <cell><m>10</m></cell>
        </row>
        <row>
            <cell halign="left"><m>g(x) </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000} </m></cell>
            <cell><m>\hphantom{000}</m></cell>
            <cell><m>\hphantom{000}</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
</exercise>
</exercisegroup>


<exercise number="35"><statement><p>
    According to the theory of relativity, the mass of an object traveling at velocity <m>v</m> is given by the function
    <me>m=\dfrac{M}{\sqrt{1-\dfrac{v^2}{c^2}}} </me>
    where <m>M</m> is the mass of the object at rest and <m>c</m> is the speed of light. Find the mass of a man traveling at a velocity of <m>0.7c</m> if his rest mass is <m>80</m> kilograms.
</p></statement>
<answer><p><m>112</m> kg</p></answer>
</exercise>

<exercise number="36"><statement><p>
    The cylinder of smallest surface area for a given volume has a radius and height both equal to <m>\sqrt[3]{\dfrac{V}{\pi}} </m>. Find the dimensions of the tin can of smallest surface area with volume <m>60</m> cubic inches.
</p></statement></exercise>

<exercise number="37"><statement><p>
    Membership in the Wildlife Society has grown according to the function
    <me>M(t) = 30t^{3/4}</me>
    where <m>t</m> is the number of years since its founding in <m>1970</m>.
    <ol label="a">
        <li><p>Sketch a graph of the function <m>M(t)</m>.</p></li>
        <li><p>What was the society's membership in <m>1990</m>?</p></li>
        <li><p>In what year will the membership be <m>810</m> people?</p></li>
    </ol>
</p></statement>
<answer><p><ol label="a">
    <li><p><image source="images/fig-ans-chap3-rev-37"  width="40%"><description>increasing concave down</description></image> </p></li>
    <li><p><m>283.7</m> or <m>\approx 284</m></p></li>
    <li><p>2051</p></li>
</ol></p></answer>
</exercise>

<exercise number="38"><statement><p>
    The heron population in Saltmarsh Refuge is estimated by conservationists at
    <me>P(t) = 360t^{-2/3}</me>
    where <m>t</m> is the number of years since the refuge was established in <m>1990</m>.
    <ol label="a">
        <li><p>Sketch a graph of the function <m>P(t)</m>.</p></li>
        <li><p>How many heron were there in <m>1995</m>?</p></li>
        <li><p>In what year will there be only <m>40</m> heron left?</p></li>
    </ol>
</p></statement></exercise>

<exercise number="39"><statement><p>
    Manufacturers of ships (and other complex products) find that the average cost of producing a ship decreases as more of those ships are produced. This relationship is called the <term>experience curve</term><idx>experience curve</idx>, given by the equation 
    <me>C = ax^{-b}</me>
    where <m>C</m> is the average cost per ship in millions of dollars and <m>x</m> is the number of ships produced. The value of the constant <m>b</m> depends on the complexity of the ship. (Source: Storch, Hammon, and Bunch, 1988)
    <ol label="a">
        <li><p>What is the significance of the constant of proportionality <m>a</m>? </p>
        <hint><p>What is the value of <m>C</m> if only one ship is built?</p></hint>
    </li>
    <li><p>For one kind of ship, <m>b = \dfrac{1}{8}</m>, and the cost of producing the first ship is <m>\$12</m> million. Write the equation for <m>C</m> as a function of <m>x</m> using radical notation.</p></li>
    <li><p>Compute the cost per ship when <m>2</m> ships have been built. By what percent does the cost per ship decrease? By what percent does the cost per ship decrease from building <m>2</m> ships to building <m>4</m> ships?</p></li>
    <li><p>By what percent does the average cost decrease from building n ships to building 2n ships? (In the shipbuilding industry, the average cost per ship usually decreases by <m>5</m> to <m>10\%</m> each time the number of ships doubles.)</p></li>
    </ol>
</p></statement>
<answer><p><ol label="a">
    <li><p>It is the cost of producing the first ship.</p></li>
    <li><p><m>C = \dfrac{12}{ \sqrt[8]{x}} </m> million</p></li>
    <li><p>About <m>\$11</m> million; about <m>8.3\%</m></p></li>
    <li><p>About <m>8.3\%</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="40"><statement><p>
    A population is in a period of <term>supergrowth</term><idx>supergrowth</idx> if its rate of growth, <m>R</m>, at any time is proportional to <m>P^k</m>, where <m>P</m> is the population at that time and <m>k</m> is a constant greater than <m>1</m>. Suppose <m>R</m> is given by
    <me>R = 0.015 P^{1.2}</me>
    where <m>P</m> is measured in thousands and <m>R</m> is measured in thousands per year.
    <ol label="a">
        <li><p>Find <m>R</m> when <m>P = 20</m>, when <m>P = 40</m>, and when <m>P = 60</m>.</p></li>
        <li><p>What will the population be when its rate of growth is <m>5000</m> per year?</p></li>
        <li><p>Graph <m>R</m> and use your graph to verify your answers to parts (a) and (b).</p></li>
    </ol>
</p></statement></exercise>


<exercisegroup cols="2"><introduction><p>Solve</p></introduction>

<exercise number="41"><statement><p><m>6t^{-3} = \dfrac{3}{500}</m></p></statement>
<answer><p><m>t=10</m></p></answer>
</exercise>

<exercise number="42"><statement><p><m>3.5 - 2.4p^{-2} = -6.1</m></p></statement></exercise>

<exercise number="43"><statement><p><m>\sqrt[3]{x+1} = 2 </m></p></statement>
<answer><p><m>x=7</m></p></answer>
</exercise>

<exercise number="44"><statement><p><m>x^{2/3}+2 = 6</m></p></statement></exercise>

<exercise number="45"><statement><p><m>(x-1)^{-3/2} = \dfrac{1}{8}</m></p></statement>
<answer><p><m>x=5</m></p></answer>
</exercise>

<exercise number="46"><statement><p><m>(2x+1)^{-1/2} =\dfrac{1}{3} </m></p></statement></exercise>

<exercise number="47"><statement><p><m>8\sqrt[4]{x+6} =24 </m></p></statement>
<answer><p><m>x=75</m></p></answer>
</exercise>

<exercise number="48"><statement><p><m>9.8 =7\sqrt[3]{z-4} </m></p></statement></exercise>

<exercise number="49"><statement><p><m>\dfrac{2}{3} (2y+1)^{0.2} = 6</m></p></statement>
<answer><p><m>y=29,524</m></p></answer>
</exercise>

<exercise number="50"><statement><p><m>1.3w^{0.3}+4.7 =5.2 </m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Solve each formula for the indicated variable.</p></introduction>

<exercise number="51"><statement><p><m>t=\sqrt{\dfrac{2v}{g}} </m>, for <m>g</m></p></statement>
<answer><p><m>g=\dfrac{2v}{t^2} </m></p></answer></exercise>

<exercise number="52"><statement><p><m>q-1=2\sqrt{\dfrac{r^2-1}{3}} </m>, for <m>r</m></p></statement></exercise>

<exercise number="53"><statement><p><m>R=\dfrac{1+\sqrt{p^2+1}}{2} </m>, for <m>p</m></p></statement>
<answer><p><m>p=\pm 2 \sqrt{R^2-R} </m></p></answer></exercise>

<exercise number="54"><statement><p><m>q=\sqrt[3]{\dfrac{1+r^2}{2}} </m>, for <m>r</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="3"><introduction><p>Simplify by applying the laws of exponents.</p></introduction>

<exercise number="55"><statement><p><m>(7t)^3 (7t)^{-1} </m></p></statement>
<answer><p><m>49t^2</m></p></answer>
</exercise>

<exercise number="56"><statement><p><m>\dfrac{36r^{-2}s}{9r^{-3}s^4} </m></p></statement>
</exercise>

<exercise number="57"><statement><p><m>\dfrac{(2k^{-1})^{-4}}{4k^{-3}} </m></p></statement>
<answer><p><m>\dfrac{k^7}{64} </m></p></answer>
</exercise>

<exercise number="58"><statement><p><m>(2w^{-3})(2w^{-3})^{5}(-5w^2) </m></p></statement>
</exercise>

<exercise number="59"><statement><p><m>\dfrac{8a^{-3/4}}{a^{-11/4}} </m></p></statement>
<answer><p><m>8a^2 </m></p></answer>
</exercise>

<exercise number="60"><statement><p><m>b^{2/3}(4b^{-2/3}-b^{1/3}) </m></p></statement>
</exercise>
</exercisegroup>


<exercise number="61"><statement><p>
    When the Concorde landed at Heathrow Airport in London, the width, <m>w</m>, of the sonic boom felt on the ground is given in kilometers by the following formula:
    <me>w=4\left(\frac{Th}{m} \right)^{1/2} </me>
    where <m>T</m> stands for the temperature on the ground in kelvins, <m>h</m> is the altitude of the Concorde when it breaks the sound barrier, and <m>m</m> is the drop in temperature for each gain in altitude of one kilometer.
    <ol label="a">
        <li><p>Find the width of the sonic boom if the ground temperature was <m>293</m> K, the altitude of the Concorde was <m>15</m> kilometers, and the temperature drop was <m>4</m> K per kilometer of altitude.</p></li>
        <li><p>Graph <m>w</m> as a function of <m>h</m> if <m>T = 293</m> and <m>m = 4</m>.</p></li>
    </ol>
</p></statement>
<answer><p><ol>
    <li><p><m>132.6</m> km</p></li>
    <li><p><image source="images/fig-ans-chap3-rev-61"  width="35%"><description>square root function</description></image> </p></li>
</ol></p></answer>
</exercise>

<exercise number="62"><statement><p>
    The manager of an office supply store must decide how many of each item in stock she should order. The Wilson lot size formula gives the most cost-efficient quantity, <m>Q</m>, as a function of the cost, <m>C</m>, of placing an order, the number of items, <m>N</m>, sold per week, and the weekly inventory cost, <m>I</m>, per item (cost of storage, maintenance, and so on).
    <me>Q=\left(\frac{2CN}{I} \right)^{1/2} </me>
    <ol label="a">
        <li><p>How many reams of computer paper should she order if she sells on average <m>80</m> reams per week, the weekly inventory cost for a ream is <m>\$0.20</m>, and the cost of ordering, including delivery charges, is <m>\$25</m>?</p></li>
        <li><p>Graph <m>Q</m> as a function of <m>N</m> if <m>C = 25</m> and <m>I = 0.2</m>.</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="63"><statement><p>
    Two businesswomen start a small company to produce saddle bags for bicycles. The number of saddle bags, <m>q</m>, they can produce depends on the amount of money, <m>m</m>, they invest and the number of hours of labor, <m>w</m>, they employ, according to the Cobb-Douglas formula
    <me>q= 0.6m^{1/4}w^{3/4} </me>
    where <m>m</m> is measured in thousands of dollars.
    <ol label="a">
        <li><p>If the businesswomen invest <m>\$100,000</m> and employ <m>1600</m> hours of labor in their first month of production, how many saddle bags can they expect to produce?</p></li>
        <li><p>With the same initial investment, how many hours of labor would they need in order to produce <m>200</m> saddle bags?</p></li>
    </ol>
</p></statement>
<answer><p><ol>
    <li><p><m>480</m></p></li>
    <li><p><m>498</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="64"><statement><p>
    A child who weighs <m>w</m> pounds and is <m>h</m> inches tall has a surface area (in square inches) given approximately by
    <me>S= 8.5 h^{0.35}w^{0.55} </me>
    <ol label="a">
        <li><p>What is the surface area of a child who weighs <m>60</m> pounds and is <m>40</m> inches tall?</p></li>
        <li><p>What is the weight of a child who is <m>50</m> inches tall and whose surface area is <m>397</m> square inches?</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="65"><statement><p>
    The cost, <m>C</m>, of insulating the ceiling in a building depends on the thickness of the insulation and the area of the ceiling. The table shows values of <m>C = f (t, A)</m>, where <m>t</m> is the thickness of the insulation and <m>A</m> is the area of the ceiling.</p><p>
    <sidebyside><tabular top="minor" halign="center"  right="minor" left="minor" bottom="minor">
        <col right="major"/>
        <row>
            <cell colspan="7">Cost of Insulation (dollars)</cell>
        </row>
        <row bottom="major">
            <cell></cell>
            <cell colspan="6">Area (sq m)</cell>
        </row>
        <row>
            <cell><line>Thickness</line><line>(cm)</line></cell>
            <cell><m>100</m></cell>
            <cell><m>200</m></cell>
            <cell><m>300</m></cell>
            <cell><m>400</m></cell>
            <cell><m>500</m></cell>
            <cell><m>600</m></cell>
        </row>
        <row>
            <cell><m>4</m></cell>
            <cell><m>72</m></cell>
            <cell><m>144</m></cell>
            <cell><m>216</m></cell>
            <cell><m>288</m></cell>
            <cell><m>300</m></cell>
            <cell><m>432</m></cell>
        </row>
        <row>
            <cell><m>5</m></cell>
            <cell><m>90</m></cell>
            <cell><m>180</m></cell>
            <cell><m>270</m></cell>
            <cell><m>360</m></cell>
            <cell><m>450</m></cell>
            <cell><m>540</m></cell>
        </row>
        <row>
            <cell><m>6</m></cell>
            <cell><m>108</m></cell>
            <cell><m>216</m></cell>
            <cell><m>324</m></cell>
            <cell><m>432</m></cell>
            <cell><m>540</m></cell>
            <cell><m>648</m></cell>
        </row>
        <row>
            <cell><m>7</m></cell>
            <cell><m>126</m></cell>
            <cell><m>252</m></cell>
            <cell><m>378</m></cell>
            <cell><m>504</m></cell>
            <cell><m>630</m></cell>
            <cell><m>756</m></cell>
        </row>
        <row>
            <cell><m>8</m></cell>
            <cell><m>144</m></cell>
            <cell><m>288</m></cell>
            <cell><m>432</m></cell>
            <cell><m>576</m></cell>
            <cell><m>720</m></cell>
            <cell><m>864</m></cell>
        </row>
        <row>
            <cell><m>9</m></cell>
            <cell><m>162</m></cell>
            <cell><m>324</m></cell>
            <cell><m>486</m></cell>
            <cell><m>648</m></cell>
            <cell><m>810</m></cell>
            <cell><m>972</m></cell>
        </row>
    </tabular></sidebyside>
    <ol label="a">
        <li><p>
            What does it cost to insulate a ceiling with an area of <m>500</m> square meters with <m>5</m> cm of insulation? Write your answer in function notation.
        </p></li>
        <li><p>Solve the equation <m>864 = f (t, 600)</m> and interpret your answer.</p></li>
        <li><p>Consider the row corresponding to a thickness of <m>4</m> cm. How does the cost of insulating the ceiling depend on the area of the ceiling?</p></li>
        <li><p>Consider the column corresponding to an area of <m>100</m> square meters. How does the cost depend on the thickness of the insulation?</p></li>
        <li><p>Given that the cost varies jointly with the thickness of the insulation and the area of the ceiling, write an equation for cost as a function of area and thickness of insulation.</p></li>
        <li><p>Use your formula from part (e) to determine the cost of insulating a building with <m>10</m> centimeters of insulation if the area of the ceiling is <m>800</m> square meters.</p></li>
    </ol>
</p></statement>
<answer><p><ol label="a">
    <li><p><m>\$450</m></p></li>
    <li><p><m>t = 8</m>: It costs <m>\$864</m> to insulate a ceiling with <m>8</m> cm of insulation over an area of <m>600</m> square meters.</p></li>
    <li><p><m>C = 0.72A</m></p></li>
    <li><p><m>C = 18T</m></p></li>
    <li><p><m>C = 0.18AT</m></p></li>
    <li><p><m>\$1440</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="66"><statement><p>
    The volume, <m>V</m>, of a quantity of helium depends on both the temperature and the pressure of the gas. The table shows values of <m>V = f (P, T )</m> for temperature in kelvins and pressure in atmospheres.</p><p>
    <sidebyside><tabular top="minor" halign="center"  right="minor" left="minor" bottom="minor">
        <col right="major"/>
        <row>
            <cell colspan="7">Volume (cubic meters)</cell>
        </row>
        <row bottom="major">
            <cell></cell>
            <cell colspan="6">Temperature (K)</cell>
        </row>
        <row>
            <cell><line>Pressure</line><line>(atmospheres)</line></cell>
            <cell><m>100</m></cell>
            <cell><m>150</m></cell>
            <cell><m>200</m></cell>
            <cell><m>250</m></cell>
            <cell><m>300</m></cell>
            <cell><m>350</m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m>18</m></cell>
            <cell><m>27</m></cell>
            <cell><m>36</m></cell>
            <cell><m>45</m></cell>
            <cell><m>54</m></cell>
            <cell><m>63</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>9</m></cell>
            <cell><m>13.5</m></cell>
            <cell><m>18</m></cell>
            <cell><m>22.5</m></cell>
            <cell><m>27</m></cell>
            <cell><m>31.5</m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m>6</m></cell>
            <cell><m>9</m></cell>
            <cell><m>12</m></cell>
            <cell><m>15</m></cell>
            <cell><m>18</m></cell>
            <cell><m>21</m></cell>
        </row>
        <row>
            <cell><m>4</m></cell>
            <cell><m>4.5</m></cell>
            <cell><m>6.75</m></cell>
            <cell><m>9</m></cell>
            <cell><m>11.25</m></cell>
            <cell><m>13.5</m></cell>
            <cell><m>15.75</m></cell>
        </row>
    </tabular></sidebyside>
    <ol label="a">
        <li><p>What is the volume of helium when the pressure is <m>4</m> atmospheres and the temperature is <m>350</m> K? Write your answer in function notation.</p></li>
        <li><p>Solve the equation <m>15 = f (3, T )</m> and interpret your answer.</p></li>
        <li><p>Consider the row corresponding to <m>2</m> atmospheres. How is the volume related to the absolute temperature?</p></li>
        <li><p>Consider the column corresponding to <m>300</m> K. How is the volume related to the pressure?</p></li>
        <li><p>Given that the volume of the gas varies directly with temperature and inversely with pressure, write an equation for volume as a function of temperature and pressure.</p></li>
        <li><p>Use your formula from part (e) to determine the volume of the helium at <m>50</m> K and pressure of <m>0.4</m> atmospheres.</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="67"><statement><p>
    In his hiking guidebook, <booktitle>Afoot and Afield in Los Angeles County</booktitle>, Jerry Schad notes that the number of people on a wilderness trail is inversely proportional to "the square of the distance and the cube of the elevation gain from the nearest road."
    <ol label="a">
        <li><p>Choose variables and write a formula for this relationship.</p></li>
        <li><p>On a sunny Saturday afternoon, you count <m>42</m> people enjoying the Rock Pool at Malibu Creek State Park. The Rock Pool is <m>1.5</m> miles from the main parking lot, and the trail includes an elevation gain of <m>250</m> feet. Calculate the constant of variation in your formula from part (a). </p>
        <hint><p>Hint: Convert the elevation gain to miles.</p></hint></li>
        <li><p>Lookout Trail leads <m>1.9</m> miles from the parking lot and involves an elevation gain of <m>500</m> feet. How many people would you expect to encounter at the end of the trail?</p></li>
    </ol>
</p></statement>
<answer><p><ol label="a">
    <li><p><m>N =\dfrac{k}{d^2E^3}</m>, where <m>N</m> is number of people, <m>d</m> is distance in miles from the road, <m>E</m> is the elevation gain, and <m>k</m> is the constant of variation.</p></li>
    <li><p><m> k\approx 0.01 </m></p></li>
    <li><p><m>3</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="68"><statement><p>
    A company's monthly production, <m>P</m>, depends on the capital, <m>C</m>, the company has invested and the amount of labor, <m>L</m>, available each month. The Cobb-Douglas model for production assumes that <m>P</m> varies jointly with <m>C^a</m> and <m>L^b</m>, where <m>a</m> and <m>b</m> are positive constants less than <m>1</m>. The Aztech Chip Company invested <m>625</m> units of capital and hired <m>256</m> workers and produces <m>8000</m> computer chips each month.
    <ol label="a">
        <li><p>Suppose that <m>a = 0.25</m>, <m>b = 0.75</m>. Find the constant of variation and a formula giving <m>P</m> in terms of <m>C</m> and <m>L</m>.</p></li>
        <li><p>If Aztech increases its labor force to <m>300</m> workers, what production level can they expect?</p></li>
        <li><p>If Aztech maintains its labor force at <m>256</m> workers, what amount of capital outlay would be required for monthly production to reach <m>16,000</m> computer chips?</p></li>
    </ol>
</p></statement>
</exercise>

</exercises>