A circles circumference is shrinking at a rate of 3/2pi cm/min. How fast is the area of the circle changing (in cm^2/min) when the circumference is 12 cm?

Answer 1

This problem can be solved in two ways. One way solves for the area in terms of the circumference then takes the derivative of the area and the circumference with respect to time. The method used here solves for the area in terms of the radius and takes the derivative of the area and radius with respect to time.

C = 2πr

r = C / 2π

A = πr²

1) dA/dt = 2πr dr/dt

2) Find r when the circumference is 12

r =12/2π = 6/π

3)Given

dC/dt = 3π/2

4)Since

dC/dt = 2π dr/dt

dr/dt = (dC/dt/)2π = 3π/2/(2π) = 3/4

5) Using 1 from above

dA/dT = 2πr dr/dt = 2π( 6/π)(3/4) = 36/4 = 9

6) The area of the circle is shrinking at a rate of 9 cm²