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Question

The volume of a sphere is increasing at the rate of 8 cm3/sec. Find the rate at which its surface area is increasing when the radius of the sphere is 12cm.

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Solution

Let r be the radius of sphere, V be the volume and S be the surface area of the sphere.

Then,

V = 43πr3and S=4πr2
dVdt=43π(3r2)drdt=4πr2drdtAlso,dVdt=8 cm3/sec8=4πr2drdtdrdt=2πr2 ...(i)
Now,
dSdt=4π(2r)drdt=8πrdrdt=8πr(2πr2) [using(i)]=16r cm2/secWhen r=12cm,dSdt=16r=1612=43=113cm2/s

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