The volume of a sphere is increasing at the rate of 3 cubic centimetre per second. Find the rate of increasing of its surface area, when the radius is 2cm.

Solution:

The formula to find the volume of a sphere is

V = 4/3 πr³

By differentiating with respect to t

dV/dt = 4/3 π 3r² dr/dt

dV/dt = 4πr² dr/dt --- (1)

It is given that

dV/dt = 3 cubic centimetre per second

r = 2 cm 

Substituting in equation (1)

3 = 4π(2)² dr/dt

dr/dt = 3/16π cm/sec

Consider S as the surface area

S = 4πr²

Differentiating with respect to t

dS/dt = 8πr dr/dt

dS/dt = 8π (2) 3/16π

dS/dt = 3 cm²/ sec

Therefore, the rate of increase of its surface area is 3 cm²/ sec.

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The volume of a sphere is increasing at the rate of 3 cubic centimetre per second. Find the rate of increasing of its surface area, when the radius is 2cm.

Summary:

The volume of a sphere is increasing at the rate of 3 cubic centimetre per second. The rate of increase of its surface area when the radius is 2cm is 3 cm²/ sec.