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Similar Question 1
<p>The following table shows the daily number of watches sold at a shop and the amount of money made from the sales.</p><img src="/qimages/278" /> <ul> <li>(a) Does the data in the table appear to follow a linear relation? Explain.</li> <li>(b) Graph the data. How does the graph compare with your hypothesis?</li> <li>(c) What is the average rate of change in revenues from <code class='latex inline'>w = 20</code> to <code class='latex inline'>w = 25</code>?</li> <li>(d) What is the cost of one watch, and how does this cost relate to the graph?</li> </ul>
Similar Question 2
<p>The purchase price, <code class='latex inline'>P</code>, of one share in company at any time, <code class='latex inline'>t</code>, in years, can be modelled by the function <code class='latex inline'>P(t) = -0.2t^3 + 2t^2 + 8t + 2, t\in[0, 13]</code>.</p><p>Determine the average rate of change of the purchase price from</p> <ul> <li>year 0 to year 5</li> <li>year 5 to year 8</li> <li>year 8 to year 10</li> <li>year 8 to year 13</li> </ul>
Similar Question 3
<p>In 1990, <code class='latex inline'>16.2\%</code> of households had a home computer, while <code class='latex inline'>66.8\%</code> of households had a home computer in <code class='latex inline'>2003</code>. Determine the average rate of change of the percent of households that had a home computer over this time period.</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>As a large snowball mets, its size changes. The volume, V, in cubic centimetres, is given by the equation <code class='latex inline'>V = \frac{4}{3}\pi r^3</code>, where r is the radius, in centimetres, and <code class='latex inline'>r \in [0, 30]</code>. The surface area, S, in square centimetres, is given by the equation <code class='latex inline'>S = 4\pi r^2</code>.</p><p>Determine the average rate of change of the surface area and the of the volume as the radius decreases from</p> <ul> <li>30 cm to 25 cm</li> <li>25 cm to 20 cm</li> </ul>
<p>The following table shows the daily number of watches sold at a shop and the amount of money made from the sales.</p><img src="/qimages/278" /> <ul> <li>(a) Does the data in the table appear to follow a linear relation? Explain.</li> <li>(b) Graph the data. How does the graph compare with your hypothesis?</li> <li>(c) What is the average rate of change in revenues from <code class='latex inline'>w = 20</code> to <code class='latex inline'>w = 25</code>?</li> <li>(d) What is the cost of one watch, and how does this cost relate to the graph?</li> </ul>
<p>A cross-country runner is training for a marathon. His training program requires him to run at different speeds for different lengths of time. His program also requires him to accelerate and decelerate at a constant rate. Today he begins by jogging for <code class='latex inline'>10</code> min at a rate of <code class='latex inline'>5</code> miles per hour. He then spends <code class='latex inline'>1</code> min accelerating to a rate of <code class='latex inline'>10</code> miles per hour. He stays at this rate for <code class='latex inline'>5</code> min. He then decelerates for <code class='latex inline'>1</code> min to a rate of <code class='latex inline'>7</code> miles per hour. He stays at this rate for <code class='latex inline'>30</code> min. Finally, to cool down, he decelerates for <code class='latex inline'>2</code> min to a rate of <code class='latex inline'>3</code> miles per hour. He stays at this rate for a final <code class='latex inline'>10</code> min and then stops.</p><p><strong>(a)</strong> What is the instantaneous rate of change in the runner&#39;s speed at <code class='latex inline'>10.5</code> min?</p><p><strong>(b)</strong> What is the runner&#39;s average rate at which he changed speeds from minute <code class='latex inline'>11</code> to minute <code class='latex inline'>49</code>?</p>
<p>A newspaper carrier delivers papers on her bicycle. She bikes to the first neighbourhood at a rate of <code class='latex inline'>10 km/hr</code>. She slows down at a constant rate over a period of <code class='latex inline'>7 s</code>, to a speed of <code class='latex inline'>5 km/hr</code>, so that she can deliver her papers. After travelling at this rate for 3 s. she sees one of her customers and decides to stop. She slows at a constant rate until she stops. It takes her 6 s to stop.</p><p>What is the average rate of change in speed from <code class='latex inline'>7</code> to <code class='latex inline'>12</code> seconds?</p>
<p>As water drains out a <code class='latex inline'>2000</code>-L hot tub, the amount of water remaining in the tub can be modelled by the function <code class='latex inline'>V = 0.000 02(100 - t)^4</code>, where <code class='latex inline'>t</code> is the time, in minutes, <code class='latex inline'>0 \leq t \leq 100</code>, and <code class='latex inline'>V(t)</code> is the volume of water, in litres, remaining in the tub at time <code class='latex inline'>t</code>.</p><p>Determine the average rate of change of the volume of water during:</p> <ul> <li>the entire 100 min</li> <li>the first 30 min</li> <li>the last 30 min</li> </ul>
<p>As a large snowball mets, its size changes. The volume, <code class='latex inline'>V</code>, in cubic centimetres, is given by the equation <code class='latex inline'>V = \frac{4}{3}\pi r^3</code>, where r is the radius, in centimetres, and <code class='latex inline'>r \in [0, 30]</code>. The surface area, S, in square centimetres, is given by the equation <code class='latex inline'>S = 4\pi r^2</code>.</p><p>Determine the average rate of change of the surface area when the surface area decreases from <code class='latex inline'>2827.43 cm^2</code> to <code class='latex inline'>1256.64 cm^2</code>.</p>
<p>In 1990, <code class='latex inline'>16.2\%</code> of households had a home computer, while <code class='latex inline'>66.8\%</code> of households had a home computer in <code class='latex inline'>2003</code>. Determine the average rate of change of the percent of households that had a home computer over this time period.</p>
<p>An investment&#39;s value, <code class='latex inline'>V(t)</code>, is modelled by the function <code class='latex inline'>V(t) = 2500(1.15)^t</code>, where <code class='latex inline'>t</code> is the number of years after funds are invested?</p><p>(a) To find the instantaneous rate of change in the value of the investment at <code class='latex inline'>t=4</code>, what intervals on either side of 4 would you choose? Why?</p><p>(b) Use your intervals from part (a) to find the instantaneous rate of change in the value of the investment at <code class='latex inline'>t=4</code>.</p>
<p>A newspaper carrier delivers papers on her bicycle. She bikes to the first neighbourhood at a rate of <code class='latex inline'>10 km/hr</code>. She slows down at a constant rate over a period of <code class='latex inline'>7 s</code>, to a speed of <code class='latex inline'>5 km/hr</code>, so that she can deliver her papers. After travelling at this rate for <code class='latex inline'>3 s</code>. she sees one of her customers and decides to stop. She slows at a constant rate until she stops. It takes her <code class='latex inline'>6 s</code> to stop.</p> <ul> <li>What is the instantaneous rate of change in speed at <code class='latex inline'>12 s</code>?</li> </ul>
<p>A newspaper carrier delivers papers on her bicycle. She bikes to the first neighbourhood at a rate of <code class='latex inline'>10 km/hr</code>. She slows down at a constant rate over a period of <code class='latex inline'>7 s</code>, to a speed of <code class='latex inline'>5 km/hr</code>, so that she can deliver her papers. After travelling at this rate for 3 s. she sees one of her customers and decides to stop. She slows at a constant rate until she stops. It takes her <code class='latex inline'>6 s</code> to stop.</p><p>(a) Draw a graph of the newspaper carrier&#39;s rate over time for the time period after she arrives the first neighbourhood.</p><p>(b) What is the average rate of change in speed over the first <code class='latex inline'>7 s</code>?</p>
<p>Vehicles lose value over time. A car is purchased for <code class='latex inline'>\$23 500</code>, but is worth only <code class='latex inline'>\$8750</code> after eight years. What is the average annual rate of change in the value of the car, as a percent?</p>
<p>If a ball is thrown into the air with a velocity of <code class='latex inline'>40 ft/sec</code>, its height after <code class='latex inline'>t</code> sec is given by <code class='latex inline'>y=40t-16t^2</code>. Find</p><p>(a) the average velocity from time <code class='latex inline'>1</code> sec to <code class='latex inline'>2 sec</code>.</p><p>(b) the instantaneous velocity at <code class='latex inline'>1 sec</code>.</p>
<p>Which of the following does not represent a situation that involves an average rate of change? Justify your answer.</p><p><strong>(a)</strong> A child grows 8 cm in 6 months.</p><p><strong>(b)</strong> The temperature at a 750 m high ski hill is <code class='latex inline'>2^{o}</code>C at the base and <code class='latex inline'>-8^{o}C</code> at the top.</p><p><strong>(c)</strong> A speedometer shows that a vehicle is travelling at 90 km/h.</p><p><strong>(d)</strong> A jogger ran 23 km in 2h.</p><p><strong>(e)</strong> The laptop cost $750.</p><p><strong>(f)</strong> A plane travelled 650 km in 3 h.</p>
<p>As a large snowball mets, its size changes. The volume, <code class='latex inline'>V</code>, in cubic centimetres, is given by the equation <code class='latex inline'>V = \frac{4}{3}\pi r^3</code>, where r is the radius, in centimetres, and <code class='latex inline'>r \in [0, 30]</code>. The surface area, S, in square centimetres, is given by the equation <code class='latex inline'>S = 4\pi r^2</code>.</p><p>Determine the average rate of change of the volume when the volume decreases from <code class='latex inline'>1675.52 cm^3</code> to <code class='latex inline'>942.48 cm^3</code>.</p>
<p>The following table gives the amount of water that is used on a farm during the first six months of the year.</p><img src="/qimages/10638" /><p><code class='latex inline'>\displaystyle \begin{array}{|l|c|} \hline Month & Volume \left(1000\right. of \left.\mathbf{m}^{3}\right) \\ \hline January & 3.00 \\ \hline February & 3.75 \\ \hline March & 3.75 \\ \hline April & 4.00 \\ \hline May & 5.10 \\ \hline June & 5.50 \\ \hline \end{array} </code></p><p>a) Plot the data in the table on a graph.</p><p>b) Find the rate of change in the volume of water used between consecutive months.</p><p>c) Between which two months is the change in the volume of water used the greatest?</p><p>d) Determine the average rate of change in the volume of water used between March and June.</p>
<p>Shelly has a cell phone plan that costs <code class='latex inline'>\$39</code> per month and allows her <code class='latex inline'>250</code> free anytime minutes. Any minutes she uses over the <code class='latex inline'>250</code> free minutes cost <code class='latex inline'>\$0.10</code> per minute. </p><p>The function</p><p><code class='latex inline'> C(m) = \begin{cases} 39, &\text{if } 0 \leq m \leq 250 \\ 0.10(m - 250)+ 39, &\text{if } m > 250 \end{cases} </code></p><p>can be used to determine her monthly cell phone bill, where <code class='latex inline'>C(m)</code> is her monthly cost in dollars and m is the number of minutes she talks. How is the average rate of change in her monthly cost change as the minutes she talks increases? Explain.</p>
<p>A company is opening a new office. The initial expense to set up the office is <code class='latex inline'>\$10 000</code>, and the company will spend another <code class='latex inline'>\$2500</code> each month in utilities until the new office opens.</p><p>(a) Write the equation that represents the company&#39;s total expenses in terms of months until the office opens.</p><p>(b) What is the average rate of change in the company&#39;s expense from <code class='latex inline'>3 \leq m \leq 6</code>?</p><p>(c) Do you expect this rate of change to vary? Why or why not?</p>
<p>A company is opening a new office. The initial expense to set up the office is <code class='latex inline'>\$10 000</code>, and the company will spend another <code class='latex inline'>\$2500</code> each month in utilities until the new office opens.</p><p><strong>a)</strong> Write the equation that represents the company’s total expenses in terms of months until the office opens.</p><p><strong>b)</strong> What is the average rate of change in the company’s expenses from <code class='latex inline'>3 \leq m \leq 6</code>?</p>
<p>The purchase price, <code class='latex inline'>P</code>, of one share in company at any time, <code class='latex inline'>t</code>, in years, can be modelled by the function <code class='latex inline'>P(t) = -0.2t^3 + 2t^2 + 8t + 2, t\in[0, 13]</code>.</p><p>Determine the average rate of change of the purchase price from</p> <ul> <li>year 0 to year 5</li> <li>year 5 to year 8</li> <li>year 8 to year 10</li> <li>year 8 to year 13</li> </ul>
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