Question

A stone is dropped into a quiet lake an waves move in circles at the speed of 5 cm /sec. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing ?

Answer 1

Area of circle, A = $\pi r^2$

$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$

Given, $\frac{dr}{dt}=5cm/sec$

$r=8cm$

Therefore,

$\frac{dA}{dt}=2\times \pi \times 8\times 5=80\pi$

Hence, area enclosed is increasing at the rate of $80\pi$ $c{m}^2/sec$.