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Question
A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
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Solution
\[\text { Let r be the radius and A be the area of the circle at any time t. Then, }\]
\[A=\pi r^2 \]
\[\Rightarrow\frac{dA}{dt}=2\pi r\frac{dr}{dt}\]
\[\Rightarrow\frac{dA}{dt}=2\pi\times4\times10\left[ \because r = 4 \text { cm and } \frac{dr}{dt} = 10 cm/\sec \right]\]
\[\Rightarrow\frac{dA}{dt} {=80\pi cm}^2 /sec\]
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