The volume of a sphere is changing at a rate of 8pi cm/sec. What is the rate of change of its surface area when the radius is 1? (the volume of a sphere is given by V=4/3pir^3 and its surface area, by A=4pi r ^2).

Answer 1

V = (4/3)π r^3
dV/dt = 4π r^2 dr/dt

given: dV/dt = 8π , r = 1
8π = 4π(1) dr/dt
dr/dt = 8π/4π = 2

A = 4πr^2 ----> (did you notice that surface area equals the derivative of volume ??)

dA/dt = 8πr dr/dt
when r = 1
dA/dt = 8π(1)(1) = 8π

the surface area is changing at 8π cm/sec