chap8-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="chap8-rev-problems"  xmlns:xi="http://www.w3.org/2001/XInclude">
    <title>Review Problems</title>

<exercisegroup cols="2"><introduction><p>Solve each system by graphing. Use the <term>ZDecimal</term> window.</p></introduction>

<exercise number="1"><statement><p><m>
    \begin{aligned}
    y \amp = -2.9x - 0.9\\
    y \amp = 1.4 - 0.6x
    \end{aligned}
</m></p></statement>
<answer><p><m>(-1, 2) </m></p></answer>
</exercise>

<exercise number="2"><statement><p><m>
    \begin{aligned}
    y \amp = 0.6x - 1.94\\
    y \amp = -1.1x + 1.29
    \end{aligned}
</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Solve each system using substitution or elimination.</p></introduction>

<exercise number="3"><statement><p><m>
    \begin{aligned}
    3x+5y \amp = 18 \\
    x-y \amp = -3
    \end{aligned}
</m></p></statement>
<answer><p><m>\left(\dfrac{1}{2}, \dfrac{7}{2} \right) </m></p></answer>
</exercise>

<exercise number="4"><statement><p><m>
    \begin{aligned}
    x+5y \amp = 11 \\
    2x+3y \amp = 8
    \end{aligned}
</m></p></statement></exercise>

<exercise number="5"><statement><p><m>
    \begin{aligned}
    \dfrac{2}{3}x - 3y \amp = 8 \\
    x + \dfrac{3}{4} y \amp = 12
    \end{aligned}
</m></p></statement>
<answer><p><m>(12, 0) </m></p></answer>
</exercise>

<exercise number="6"><statement><p><m>
    \begin{aligned}
    3x \amp = 5y-6 \\
    3y \amp = 10-11x
    \end{aligned}
</m></p></statement></exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Decide whether each system is inconsistent, dependent, or consistent and independent.</p></introduction>

<exercise number="7"><statement><p><m>
    \begin{aligned}
    2x -3y \amp = 4 \\
    x+ 2y \amp = 7
    \end{aligned}
</m></p></statement>
<answer><p>Consistent and independent</p></answer>
</exercise>

<exercise number="8"><statement><p><m>
    \begin{aligned}
    2x -3y \amp = 4 \\
    6x -9y \amp = 4
    \end{aligned}
</m></p></statement>
</exercise>

<exercise number="9"><statement><p><m>
    \begin{aligned}
    2x -3y \amp = 4 \\
    6x -9y \amp = 12
    \end{aligned}
</m></p></statement>
<answer><p>Dependent</p></answer>
</exercise>

<exercise number="10"><statement><p><m>
    \begin{aligned}
    x -y \amp = 6 \\
    x + y \amp = 6
    \end{aligned}
</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Solve each system using Gaussian reduction.</p></introduction>

<exercise number="11"><statement><p><m>
\begin{alignedat}{5}
    x \amp {}+{} \amp 3y \amp {}-{} \amp z\amp {}={}  3 \\
    2x \amp {}-{} \amp  y \amp {}+{} \amp  3z \amp {}={}   1 \\
    3x \amp {}+{} \amp 2y \amp {}+{} \amp  z \amp {}={}  5
\end{alignedat} </m></p></statement>
<answer><p><m>(2, 0, -1)</m></p></answer>
</exercise>

<exercise number="12"><statement><p><m>
\begin{alignedat}{5}
    x \amp {}+{} \amp y \amp {}+{} \amp z\amp {}={}  2 \\
    3x \amp {}-{} \amp  y \amp {}+{} \amp  z \amp {}={}   4 \\
    2x \amp {}+{} \amp y \amp {}+{} \amp 2z \amp {}={}  3
\end{alignedat} </m></p></statement>
</exercise>

<exercise number="13"><statement><p><m>
    \begin{aligned}
    x +z \amp = 5 \\
    y-z \amp = -3 \\
    2x + z \amp = 7
    \end{aligned} </m></p></statement>
<answer><p><m>(2, -5, 3)</m></p></answer>
</exercise>

<exercise number="14"><statement><p><m>
\begin{alignedat}{5}
    x \amp {}+{} \amp 4y \amp {}+{} \amp 4z\amp {}={}  0 \\
    3x \amp {}+{} \amp 2y \amp {}+{} \amp  z \amp {}={}   -4 \\
    2x \amp {}-{} \amp 4y \amp {}+{} \amp z \amp {}={}  -11
\end{alignedat} </m></p></statement>
</exercise>

<exercise number="15"><statement><p><m>
\begin{alignedat}{5}
    \dfrac{1}{2} x \amp {}+{} \amp y \amp {}+{} \amp z\amp {}={} 3 \\
    x \amp {}-{} \amp 2y \amp {}-{} \amp \dfrac{1}{3}z \amp {}={} -5 \\
    \dfrac{1}{2} x \amp {}-{} \amp 3y \amp {}-{} \amp \dfrac{2}{3} z \amp {}={} -6
\end{alignedat} </m></p></statement>
<answer><p><m>(-2, 1, 3)</m></p></answer>
</exercise>

<exercise number="16"><statement><p><m>
\begin{alignedat}{5}
    \dfrac{3}{4} x \amp {}-{} \amp \dfrac{1}{2}y \amp {}+{} \amp 6z\amp {}={} 2 \\
    \dfrac{1}{2}x \amp {}+{} \amp y \amp {}-{} \amp \dfrac{3}{4}z \amp {}={} 0 \\
    \dfrac{1}{4} x \amp {}+{} \amp \dfrac{1}{2}y \amp {}-{} \amp \dfrac{1}{2} z \amp {}={} 0
\end{alignedat} </m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Use matrix reduction to solve each system.</p></introduction>

<exercise number="17"><statement><p><m>
\begin{alignedat}{3}
     x \amp {}-{} \amp 2y \amp {}={} 5 \\
    2x \amp {}+{} \amp y  \amp {}={} 5 
\end{alignedat} 
</m></p></statement>
<answer><p><m>(3,-1)</m></p></answer>
</exercise>

<exercise number="18"><statement><p><m>
\begin{alignedat}{3}
    4x \amp {}-{} \amp 3y \amp {}={} 16 \\
    2x \amp {}+{} \amp y  \amp {}={} 8 
\end{alignedat} 
</m></p></statement>
</exercise>

<exercise number="19"><statement><p><m>
\begin{alignedat}{3}
    2x \amp {}-{} \amp y \amp {}={} 7 \\
    3x \amp {}+{} \amp 2y  \amp {}={} 14 
\end{alignedat}  </m></p></statement>
<answer><p><m>(4, 1)</m></p></answer>
</exercise>

<exercise number="20"><statement><p><m>
\begin{alignedat}{5}
    2x \amp {}-{} \amp y \amp {}+{} \amp 3z\amp {}={}  -6 \\
    x \amp {}+{} \amp 2y \amp {}-{} \amp  z \amp {}={}   7 \\
    3x \amp {}+{} \amp y \amp {}+{} \amp z \amp {}={}  2
\end{alignedat} </m></p></statement>
</exercise>

<exercise number="21"><statement><p><m>
\begin{alignedat}{5}
    x \amp {}+{} \amp 2y \amp {}-{} \amp z\amp {}={} -3\\
    2x \amp {}-{}\amp 3y \amp {}+{} \amp 2z\amp {}={} 2 \\
    x \amp {}-{} \amp y \amp {}+{} \amp 4z \amp {}={} 7
\end{alignedat} </m></p></statement>
<answer><p><m>(4, 1)</m></p></answer>
</exercise>

<exercise number="22"><statement><p><m>
\begin{alignedat}{5}
    x \amp {}+{} \amp y \amp {}+{} \amp z\amp {}={} 1 \\
    2x \amp {}-{} \amp y \amp {}-{} \amp z \amp {}={} 2 \\
    2x \amp {}-{} \amp y \amp {}+{} \amp 3z \amp {}={} 2
\end{alignedat} </m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Solve the system by finding the reduced row echelon form of the augmented matrix.</p></introduction>

<exercise number="23"><statement><p><m>
\begin{alignedat}{7}
    2a \amp{}+{}\amp 3b \amp{}-{}\amp 4c\amp{}-{}\amp 5d\amp {}={} 3 \\
    2a \amp{}-{}\amp 3b \amp{}+{}\amp 4c\amp{}-{}\amp 7d\amp {}={} -11 \\
    3a \amp{}+{}\amp b \amp{}-{}\amp 8c\amp{}+{}\amp d\amp {}={} 9 \\
    4a \amp{}-{}\amp 7b \amp{}-{}\amp 5c\amp{}+{}\amp 3d\amp {}={} -4 
\end{alignedat} 
</m></p></statement>
<answer><p><m>(4, 3, 1, 2)</m></p></answer>
</exercise>

<exercise number="24"><statement><p><m>
\begin{alignedat}{7}
    -a \amp{}-{}\amp 2b \amp{}+{}\amp 5c\amp{}+{}\amp 2d\amp {}={} 15 \\
    -2a \amp{}+{}\amp 3b \amp{}+{}\amp 2c\amp{}+{}\amp d\amp {}={} 15 \\
    2a \amp{}-{}\amp 4b \amp{}+{}\amp 6c\amp{}+{}\amp 9d\amp {}={} 20 \\
    6a \amp{}+{}\amp 8b \amp{}+{}\amp 7c\amp{}-{}\amp 2d\amp {}={} 0 
\end{alignedat} 
</m></p></statement>
</exercise>

<exercise number="25"><statement><p><m>
\begin{alignedat}{7}
    2a \amp{}-{}\amp b \amp{}-{}\amp 3c\amp{}+{}\amp d\amp{}+{}\amp 5e\amp {}={} 7 \\
    4a \amp{}+{}\amp 6b \amp{}-{}\amp 3c\amp{}-{}\amp d\amp{}+{}\amp e\amp {}={} -6 \\
    5a \amp{}+{}\amp 2b \amp{}-{}\amp 9c\amp{}-{}\amp 4d\amp{}+{}\amp 7e\amp {}={} 3 \\
    6a \amp{}-{}\amp 2b \amp{}+{}\amp 7c\amp{}+{}\amp 2d\amp{}-{}\amp 8e\amp {}={} 11 \\
    7a \amp{}+{}\amp 8b \amp{}+{}\amp 2c\amp{}+{}\amp 6d\amp{}+{}\amp e\amp {}={} 2 
\end{alignedat} 
</m></p></statement>
<answer><p><m>(2,-1, 5,-3, 4)</m></p></answer>
</exercise>

<exercise number="26"><statement><p><m>
\begin{alignedat}{7}
    a \amp{}-{}\amp 4b \amp{}+{}\amp 2c\amp{}+{}\amp 3d\amp{}-{}\amp e\amp {}={} 7 \\
    a \amp{}+{}\amp 2b \amp{}-{}\amp 5c\amp{}+{}\amp 2d\amp{}-{}\amp 3e\amp {}={} -6 \\
    3a \amp{}+{}\amp 3b \amp{}+{}\amp 2c\amp{}-{}\amp 4d\amp{}+{}\amp 3e\amp {}={} 3 \\
    -2a \amp{}-{}\amp 3b \amp{}+{}\amp 5c\amp{}+{}\amp 4d\amp{}-{}\amp e\amp {}={} 11 \\
    -4a \amp{}+{}\amp 3b \amp{}-{}\amp c\amp{}+{}\amp d\amp{}+{}\amp 2e\amp {}={} 2 
\end{alignedat} 
</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup><introduction><p>For Problems 27<ndash/>32, solve by writing and solving a system of linear equations in two or three variables.</p></introduction>

<exercise number="27"><statement><p>A math contest exam has 40 questions. A contestant scores <m>5</m> points for each correct answer but loses <m>2</m> points for each wrong answer. Lupe answered all the questions and her score was <m>102</m>. How many questions did she answer correctly?</p></statement>
<answer><p><m>26</m></p></answer>
</exercise>

<exercise number="28"><statement><p>A game show contestant wins <m>\$25</m> for each correct answer he gives but loses <m>\$10</m> for each incorrect response. Roger answered <m>24</m> questions and won <m>\$355</m>.
How many answers did he get right?</p></statement>
</exercise>

<exercise number="29"><statement><p>Barbara wants to earn <m>\$500</m> a year by investing <m>\$5000</m> in two accounts, a savings plan that pays <m>8\%</m> annual interest and a high-risk option that pays <m>13.5\%</m> interest. How much should she invest in each account?</p></statement>
<answer><p><m>\$3181.82</m> at <m>8\%</m>, <m>\$1818.18</m> at <m>13.5\%</m></p></answer>
</exercise>

<exercise number="30"><statement><p>An investment broker promises his client a <m>12\%</m> return on her funds. If the broker invests <m>$3000</m> in bonds paying <m>8\%</m> interest, how much must he invest in stocks paying <m>15\%</m> interest to keep his promise?</p></statement>
</exercise>

<exercise number="31"><statement><p>The perimeter of a triangle is <m>30</m> centimeters. The length of one side is <m>7</m> centimeters shorter than the second side,
and the third side is <m>1</m> centimeter longer than the second side. Find the length of each side.</p></statement>
<answer><p><m>5</m> cm, <m>12</m> cm, <m>13</m> cm</p></answer>
</exercise>

<exercise number="32"><statement><p>A company ships its product to three cities: Boston, Chicago, and Los Angeles. The cost of shipping is <m>\$10</m> per crate to Boston, <m>\$5</m> per crate to Chicago, and <m>\$12</m> per crate to Los Angeles. The company's shipping budget
for April is <m>\$445</m>. It has <m>55</m> crates to ship, and demand for its product is twice as high in Boston as in Los Angeles. How many crates should the company ship to each destination?</p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Graph each inequality.</p></introduction>

<exercise number="33"><statement><p><m>3x - 4y\lt 12</m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-33"  width="50%"><description>inequality in two variables</description></image> </p></answer>
</exercise>

<exercise number="34"><statement><p><m>x \gt 3y - 6</m></p></statement>
</exercise>

<exercise number="35"><statement><p><m>y\lt -\dfrac{1}{2}</m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-35"  width="50%"><description>inequality in two variables</description></image> </p></answer>
</exercise>

<exercise number="36"><statement><p><m>-4\le x \lt 2</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>Graph the solutions to each system of inequalities.</p></introduction>

<exercise number="37"><statement><p><m>y\gt 3, ~x \le 2</m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-37"  width="50%"><description>system of inequalities</description></image> </p></answer>
</exercise>

<exercise number="38"><statement><p><m>y \ge x, ~x \gt 2</m></p></statement>
</exercise>

<exercise number="39"><statement><p><m>3x - y \lt 6, ~x + 2y \gt 6</m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-39"  width="50%"><description>system of inequalities</description></image> </p></answer>
</exercise>

<exercise number="40"><statement><p><m>x - 3y \gt 3, ~y \lt x + 2</m></p></statement>
</exercise>
</exercisegroup>


<exercisegroup cols="2"><introduction><p>For Problems 41<ndash/>44,
<ol label="a">
    <li><p>Graph the solutions to the system of inequalities.</p></li>
    <li><p>Find the coordinates of the vertices.</p></li>
</ol></p></introduction>

<exercise number="41"><statement><p><m>
    \begin{aligned}
        3x - 4y \le 12\\
        x \ge 0, ~y \le 0
    \end{aligned}
    </m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-41"  width="45%"><description>system of inequalities</description></image> </p></answer>
</exercise>

<exercise number="42"><statement><p><m>
    \begin{aligned}
        2x - y \le 6\\
        y \le x \\
        x \ge 0, ~y \ge 0
    \end{aligned}
    </m></p></statement>
</exercise>

<exercise number="43"><statement><p><m>
    \begin{aligned}\\
        x+y \le 5\\
        y \ge x \\
        y \ge 2, ~x \ge 0
    \end{aligned}
    </m></p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-43"  width="45%"><description>system of inequalities</description></image> </p></answer>
</exercise>

<exercise number="44"><statement><p><m>
    \begin{aligned}\\
        x-y \le -3\\
        x+y\le 6\\
        x \le 4 \\
        x \ge 0, ~y \ge 0
    \end{aligned}
    </m></p></statement>
</exercise>
</exercisegroup>


<exercise number="45"><statement><p>Ruth wants to provide cookies for the customers at her video rental store. It takes <m>20</m> minutes to mix the ingredients for each batch of peanut butter cookies and <m>10</m> minutes to bake them. Each batch of granola cookies takes <m>8</m> minutes to mix and <m>10</m> minutes to bake. Ruth does not want to use the oven more than <m>2</m> hours a day or
to spend more than <m>2</m> hours a day mixing ingredients. Write a system of inequalities for the number of batches of peanut butter cookies and of granola cookies that Ruth can make in one day and graph the solutions.</p></statement>
<answer><p><image source="images/fig-ans-chap8-rev-45"  width="40%"><description>system of inequalities</description></image></p><p><m>20p + 8g \le 120, ~10p + 10g \le 120</m></p></answer>
</exercise>

<exercise number="46"><statement><p>A vegetarian recipe calls for no more than <m>32</m> ounces of a combination of tofu and tempeh. Tofu provides <m>2</m> grams of protein per ounce and tempeh provides <m>1.6</m> grams of protein per ounce. Graham would like the dish to provide at least <m>56</m> grams of protein. Write a system of inequalities for the amount of tofu and the amount of tempeh for the recipe and graph the solutions.</p></statement>
</exercise>


<exercisegroup><introduction><p>For Problems 47<ndash/>48,
<ol label="a">
    <li><p>Graph the set of feasible solutions.</p></li>
    <li><p>Find the vertex that gives the minimum of the objective function, and find the minimum value.</p></li>
    <li><p>Find the vertex that gives the maximum of the objective function, and find the maximum value.</p></li>
</ol></p></introduction>

<exercise number="47"><statement><p>Objective function <m>C = 18x + 48y</m> with constraints <m>x \ge 0</m>, <m>y \ge 0</m>, <m>3x + y \ge 3</m>, <m>2x + y \le 12</m>, <m>x + 5y \le 15</m></p></statement>
<answer><p>(b) <m>(1, 0)</m>; <m>18\hphantom{00000}</m>  (c) <m>(5, 2)</m>; <m>186</m></p></answer>
</exercise>

<exercise number="48"><statement><p>Objective function <m>C = 10x - 8y</m> with constraints <m>x \ge 0</m>, <m>y \ge 0</m>, <m>5x - y \ge 2</m>, <m>x + 2y \le 18</m>, <m>x -y \le 3</m></p></statement>
</exercise>
</exercisegroup>


<exercise number="49"><statement><p>Ruth wants to provide cookies for the customers at her video rental store. It takes <m>20</m> minutes to mix the ingredients for each batch of peanut butter cookies and <m>10</m> minutes to bake them. Each batch of granola cookies takes <m>8</m> minutes to mix and <m>10</m> minutes to bake. Ruth does not want to use the oven more than <m>2</m> hours a day or
to spend more than <m>2</m> hours a day mixing ingredients.
<ol>
    <li><p>Write a system of inequalities for the number of batches of peanut butter cookies and granola cookies Ruth can make in one day and graph the solutions.</p></li>
    <li><p>Ruth decides to sell the cookies. If she charges <m>25</m>¢ per peanut butter cookie and <m>20</m>¢ per granola cookie, she will sell all the cookies she bakes. Each batch contains <m>50</m> cookies. How many batches of each type of cookie should she bake to maximize her income? What is the maximum income?</p></li>
</ol></p></statement>
<answer><p><ol>
    <li><p><m>20p + 8g\le 120</m>, <m>10p + 10g \le 120</m>, <m>p\ge 0</m>, <m>g \ge 0</m></p><p><image source="images/fig-ans-chap8-rev-45"  width="40%"><description>system of inequalities</description></image></p></li>
    <li><p><m>2</m> batches peanut butter cookies; <m>10</m> batches of granola cookies; for <m>\$125</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="50"><statement><p>A vegetarian recipe calls for <m>32</m> ounces of a combination of tofu and tempeh. Tofu provides <m>2</m> grams of protein per ounce and tempeh provides <m>1.6</m> grams of protein per ounce. Graham would like the dish to provide at least <m>56</m> grams of protein.
<ol>
    <li><p>Write a system of inequalities for the amount of tofu and the amount of tempeh for the recipe and graph the solutions.</p></li>
    <li><p>Suppose that tofu costs <m>12</m>¢ per ounce and tempeh costs <m>16</m>¢ per ounce. What is the least expensive combination of tofu and tempeh Graham can use for the recipe? How much will it cost?</p></li>
</ol></p></statement>
</exercise>
</exercises>