Question

A stone is dropped into a quiet lake and waves move in a circle at a speed of $3.5$ cm/sec. At the instant when the radius of the circular wave is $7.5$ cm, how fast is the enclosed area increasing ?

Answer 1

Let r be the radius and A be the area of the circular wave at any time t. Then,

$A=\pi r^2$ and $\frac{dr}{dt}=3.5cm/sec.$

Now, $A=\pi r^2$

$\frac{dA}{dt}=\pi \left(2r\frac{dr}{dt}\right)=2\pi r\frac{dr}{dt}$

$\frac{dA}{dt}=2\pi r\left(3.5\right)=7\pi r$

$(\frac{dA}{dt}{)}_{r=7.5}=7\pi (7.5)=52.5\pi c{m}^2/sec.$