A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant when the radius of the circular waves is 10 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.
$therefore space space space space space straight A space equals space πr squared space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space dA over dt space equals space straight pi space. space space 2 straight r dr over dt space equals space 2 πr dr over dt$
From given condition, $dr over dt space equals space 4 space cm divided by sec$
$therefore space space space space space when space straight r space equals space 10 space cm comma space space space space space dA over dt space equals 2 space straight pi space cross times space 10 space cross times space 4 space cm squared divided by straight s space equals space 80 space cm squared divided by straight s therefore space space space space the space enclosed space area space is space increasing space at space the space rate space of space 80 space cm squared divided by straight s comma space space when space space straight r space equals space 10 space cm.$

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Application Of Derivatives

A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant when the radius of the circular waves is 10 cm, how fast is the enclosed area increasing?

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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