Question

The hypotenuse AB of a right triangle ABC is 5 ft, and one leg, AC, is decreasing at the rate of 2 ft/sec. The rate, in square feet per second, at which the area is changing when AC = 3 is?

Answer 1

The hypotenuse AB is 5 and the length of one leg AC is x, then by the Pythagorean Theorem, the length of the other leg BC is sqrt(25 - x^2). So the area of the triangle is

A = (1/2)x sqrt(25 - x^2)
dA/dt = {(25-2x^2)/[2 sqrt(25-x^2)]} dx/dt

Since dx/dt = -2, it follows that when x = 3,
dA/dt = {(25 - 18)/[2 sqrt(25-9)]} (-3)
= -(-21)/8