<p>The length of a rectangle is increasing at a rate of 5 cm/s and its width is increasing at a rate of 4 cm/s. When the length is 10 cm and the width is 12 cm, how fast is the area of the rectangle increasing?</p>
<p>In general, with related rates, you want to do the following:</p>
<p>(1) Set up a formula that relates the variables. In this case, the formula that you want is A = l * w.</p>
<p>(2) Take note of the information that you have. In this case, you know that l = 10, w = 12, dl/dt = +5, and dw/dt = 4. The question, asking the rate at which the area is increasing, is asking for you to find dA/dt.</p>
<p>(3) Take the derivative (implicitly) of your formula equation with respect to time (t). In this particular case, remember that you need the Product Rule.</p>
<p>(4) Substitute the values that you know, solving for the value that you need. Sometimes (not in this case), you will need additional information that you don't currently have but that you can find (i.e. the hypotenuse of a right triangle, given the two legs), and you'll need to do that here first.</p>
<p>(5) Remember your units. In this case, the units for area are cm^2, and the time units are still in seconds, so your units for this answer should be in cm^2/sec.</p>
<p>If you want to check your answer, the answer I got for this question is 100 cm^2/sec. I've left some of these calculations for you, however.</p>
<p>This is an easy problem. Start with listing all the known and unknown quantities and with the formula for the area of a rectangle:</p>
<p>We know:</p>
<p>Length: 10 cm
Width: 12 cm
Change in length (dl/dt): 5 cm/s
Change in width (dw/dt): 4 cm/s</p>
<p>We want to find:</p>
<p>Change in Area over time (dA/dt)</p>
<p>Here's the solution to the problem:</p>
<p>A = l*w</p>
<p>(dA/dt) = l*(dw/dt) + w(dl/dt)<br>
[implicitly differentiate both sides of the equation with respect to time "t"]</p>
<p>(dA/dt) = 10<em>4 + 12</em>5
[Plug-in all quantities given in the original problem.]</p>
<p>(dA/dt) = 40 + 60 = 100 cm^2/s</p>
<p>The area of the rectangle is changing at a rate of 100 cm^2/s.</p>
<p>Here are some guidelines for solving related-rate problems.</p>
<ol>
<li>Identify all given quantities and quantities to be determined. Make a sketch and label the quantities.</li>
<li>Write an equation involving the variables whose rates of change either are given or are to be determined.</li>
<li>Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time "t".</li>
<li>After completing Step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.</li>
</ol>
<p>The math should be correct. Hope this helps.</p>
<p>thank you guys.. wow this was so easy</p>
<p>What did you forget to do? Did you forget to use the product rule?</p>