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?

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{A}}\text{be the area of the circle at any time}\mathit{\text{t}}\text{. Then,} \\ \mathit{\text{A}}\text{=π}{\mathit{\text{r}}}^2 \\ \text{⇒}\frac{dA}{dt}\text{=2π}\mathit{\text{r}}\frac{dr}{dt} \\ \text{⇒}\frac{dA}{dt}\text{=2π×4×10}\left[\because r=4\mathrm{cm}\mathrm{and}\frac{dr}{dt}=10\mathrm{cm}/\sec \right] \\ \text{⇒}\frac{dA}{dt}{\text{=80π cm}}^2\text{/sec} \end{gathered}$$