A stone is dropped into a quiet lake. If the waves moves in circles at the rate of 30 cm/sec when the radius is 50 m, the rate of increase of enclosed area, is 

Let the radius of the circular wave ring be r cm at any time t.

 Then $\frac{dr}{dt}=30\mathrm{cm}/\sec$ (Given)

Let A be the area of the enclosed ring. Then, 

$\begin{array}{l}A=\pi r^2\\ \Rightarrow \frac{dA}{dt}=2\pi r\frac{dr}{dt}\\ \Rightarrow \frac{dA}{dt}=2\pi \times 50\times \frac{30}{100}{\mathrm{m}}^2/\sec =30\pi {\mathrm{m}}^2/\sec \end{array}$