Subjective Type

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

The area of a circle $$(A)$$ with radius $$(r)$$ is given by $$A=\pi r^2$$.

Therefore, the rate of change of area $$(A)$$ with respect to time $$(t)$$ is given by,

$$\displaystyle \frac {dA}{dt}=\frac{d}{dt}(\pi r^2)=\frac{d}{dr}(\pi r^2)\frac{dr}{dt}=2\pi r \frac{dr}{dt},$$ [By chain rule]

It is given that $$\displaystyle \frac {dr}{dt}=5 cm$$

Thus, when $$r = 8 cm,$$

$$\displaystyle \frac {dA}{dt}=2 \pi(8)(5)=80 \pi cm^2/s$$

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