If the radius of a sphere is increasing at a rate of 3 centimeters per second, how fast is the volume?

Question

increasing when the radius is 9 centimeters?


A. 108pi cm^3/s
B. 972pi cm^3/s
C. 486pi cm^3/s
D. 972 cm^3/s

Please show me how to get to the answer, and not just the answer. Thank you.

Answer 1

Let Volume of sphere = V

The formula for volume of sphere:
V=4pi (r^3) /3

Differentiate this equation w/ respect to time (t)
dV/dt = 4 pi r ^2 dr/dt

We are given the following:
dr/dt= 3
r=9

Substitute into equation above:
dV/dt = 4 pi (9^2) 3=
4(pi)(81)(3)= 972 pi cm^3/sec

Answer = B

Answer 2

V=4/3*pi*r^3
dV/dt=dV/dr*dr/dt= 4*pi*r^2 *3=4*pi *81*3=972*pi cm^3/s