The radius of an air-bubble is increasing at the rate of $1 half cm space per space second.$ At what rate is the volume of the bubble increasing when the radius is 1 cm?

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Solution

Let V be the volume of the air-bubble whose radius is r at any time t.

therefore space space space space space space space straight V space equals space 4 over 3 πr cubed space space space space space space space space space space space space space space space space space rightwards double arrow space space space space dV over dt space equals space 4 over 3 straight pi cross times 3 straight r squared dr over dt equals space 4 πr squared dr over dt

But

dr over dt space equals 1 half cm divided by sec.

therefore space space space space dV over dt space equals space 4 πr squared space cross times space 1 half space equals space 2 space πr squared

When

straight r space equals 1 comma space space space space dV over dt space equals space 2 straight pi space cross times space left parenthesis 1 right parenthesis squared space equals space 2 space straight pi

therefore space space space space space rate space of space increase space of space volume space space equals space 2 space straight pi space cubic space cm divided by sec.

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