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A right triangle has a fixed hypotenuse of length units. If a leg is increasing in length at a rate of unit per second, find the rate of change in area of the triangle when that same leg is  units long.

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

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

So start with an area equation: .

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

Use the Pythagorean Theorem to find the length of the height in terms of .
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.

Implicitly differentiate with respect to time to find a function for ,

the rate that the area is changing.

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

If you did it right, the only variable that will remain is .