The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 60 mm? (Round your answer to two decimal places.)

Question

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing (in mm³/s) when the diameter is 60 mm? (Round your answer to two decimal places.)

Answer 1

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing (in mm³/s) when the diameter is 60 mm? (Round your answer to two decimal places.)

Detailed Solution:

Given/Known: dv/dt = 3 mm/s; d = 60 mm; r = 30 mm; v = (4/3)πr³; dv/dt = ?? mm³/s

v = (4/3)πr³ ==> d/dt (v) = d/dt ( (4/3)πr³ ) ==> dv/dt = (4/3)π(3)r² (dr/dt) ==> dv/dt = 4πr² (dr/dt)

dv/dt = 4πr² (dr/dt)

dv/dt = 4π(30)² (3) = 10800π mm³/s

Answer 2

V = 4/3πr³

dV/dt = 4πr²·dr/dt

dV/dt = 10,800π mm³/sec