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?
Solution:
Let r be the radius and A be the area of the circle at any time t. Then,
$A=\pi r^{2}$
$\Rightarrow \frac{d A}{d t}=2 \pi r \frac{d r}{d t}$
$\Rightarrow \frac{d A}{d t}=2 \pi \times 4 \times 10$ $\left[\because r=4 \mathrm{~cm}\right.$ and $\left.\frac{d r}{d t}=10 \mathrm{~cm} / \mathrm{sec}\right]$
$\Rightarrow \frac{d A}{d t}=80 \pi \mathrm{cm}^{2} / \mathrm{sec}$