related rates Problem 15

Question

The altitude of a triangle is increasing at a rate of 1.5 centimeters/minute while the area of the triangle is increasing at a rate of 5 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 7 centimeters and the area is 96 square centimeters?

Answer 1

Let A = area of triangle at time t, b = length of base at time t, and h = length of altitude at time t.

Given: dh/dt = 1.5 and dA/dt = 5.

Find db/dt when h = 7 and A = 96.

A = (1/2)bh

So, dA/dt = (1/2)[b(dh/dt) + h(db/dt)]

5 = (1/2)[1.5b + 7(db/dt)]

When h = 7 and A = 96, we have 96 = (1/2)(7b). So, b = 27.43.

Therefore, 5 = (1/2)[41.14 + 7(db/dt)]

10 = 41.14 + 7(db/dt)

db/dt = -4.45 cm/min (the base is decreasing at the rate of 4.45 cm/min when h = 7 cm and A = 96 cm².