CPM Homework Help : CALC3RD Problem 7-32

A right triangle has a fixed hypotenuse of length $13$ units. If a leg is increasing in length at a rate of $\frac { 1 } { 2 }$ unit per second, find the rate of change in area of the triangle when that same leg is $5$ units long.

Step 1:

Sketch the triangle and label the known information. Right triangle, vertical leg labeled, h = height, hypotenuse labeled, c = hypotenuse = 13.

Step 2:

The question is asking about the rate-of-change of area.

So start with an area equation: $A=\frac{1}{2}bh$ .

Notice that there are two variables. It would be easier to have only one.

Step 3:

Use the Pythagorean Theorem to find the length of the height in terms of $b$ .
Then rewrite the function with only one variable: Right triangle, vertical leg labeled, h = square root of the quantity 13 squared minus b squared, hypotenuse labeled, c = 13.

$A=\frac{1}{2}(b)(\sqrt{13^{2}-b^{2}})$

Step 4:

Implicitly differentiate with respect to time to find a function for $\frac{dA}{dt}$ ,

the rate that the area is changing.

Step 5:

Substitute in all values that you know, including the specific base length, $5$ .

If you did it right, the only variable that will remain is $\frac{dA}{dt}$ .