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Question

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?

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Solution

Let r be the radius

Given, Waves move in a circle at speed of 5 cm/s

i.e., Radius of circle increasing at a rate of 5 cm/s

drdt=5 cm/s

Area of circle, A=πr2

Now,

dAdt=d(πr2)dt

=πd(r2)dt

=2πrdrdt

=10πr[drdt=5]

When r=8 cm

dAdt=10π×8=80π cm2/s

Hence, enclosed area is increasing at the rate of 80π cm2/s when radius is 8 cm.



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