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Question

A balloon which always remains spherical on inflation, is being inflated pumping in 900 cubic cm of gas per second.  Find the rate at which the radius of the balloon increases when the radius is 15 cm.


Solution

At any instant time t let the radius of the balloon be r and its volume be V, then Volume of balloon V=(43)πr3
The balloon is being inflated at 900 cubic cm/s i.e., the rate of change of volume with respect to time is 900cm3/s.
On differentiating w.r.t. t, we get
Rate of change of volume dVdt=(43π)(3r2drdt)
Given r=15cm900=(43π){3(15)2drdt}drdt=9003×(15)2×34π
Rate of change of radius r, drdt=9004π×(15)2=225π×225=1πcm/s
Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is 1π cm/s.


Mathematics

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