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Application Of Derivatives

Question

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A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 Cm, how fast is the enclosed area increasing?

Solution

Let r be the radius of the circular wave and A the enclosed area at time t.

therefore space space space space space space straight A space equals space πr squared space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space space dA over dt space equals space straight pi space. space 2 straight r space dr over dt space equals space 2 space πr space dr over dt

From given condition,

dr over dt space equals space 5 space cm divided by sec

therefore space space space space space space space when space straight r space equals space 8 space cm comma space space space dA over dt space equals space 2 straight pi cross times 8 cross times 5 space cm squared divided by straight s space equals space 80 space cm squared divided by straight s therefore space space space space the space enclosed space area space is space increasing space at space the space rate space of space 80 space cm squared divided by straight s comma space space when space straight r space equals space 8 space cm.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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