At what rate is the base of the triangle changing when the altitude is 12.00012.000 centimeters and the area is 87.00087.000 square centimeters?

Question

The altitude of a triangle is increasing at a rate of 2.0002.000 centimeters/minute while the area of the triangle is increasing at a rate of 2.0002.000 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 12.00012.000 centimeters and the area is 87.00087.000 square centimeters?  

Answer 1

Area = (1/2) * Base * Height

where A is the area, B is the base, and H is the height/altitude,

all of which are functions of time t...

dH/dt = 2 cm/min

dA/dt = 2 cm^2/min

Product rule says:

dA/dt = (1/2) [B * dH/dt + dB/dt * H]

A=87 when H=12 --> 87 = (1/2)*B*12

87 = 6B

B = 87/6

2 = (1/2) [ 87/6 * 2 + dB/dt * 12]

4 = [ 87/3 + dB/dt * 12]

12/3 = 87/3 + dB/dt * 12

-75/3 = dB/dt * 12

-25 = dB/dt * 12

-25/12 = dB/dt

-2 and 1/12 = dB/dt

So when the height increases at 2 cm/min, the

base DECREASES at a rate of -2 and 1/12 cm/min

at the instant the area is 87