CPM Homework Help : CALC Problem 7-33

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.

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

Step 4:

the rate that the area is changing.

Step 5:

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