Question

The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
(a) π cm²/sec
(b) 2π cm²/sec
(c) π² cm²/sec
(d) 2π² cm²/sec²

Answer 1

(c) π² cm²/sec

$$\begin{gathered} \text{Let}\mathit{\text{D}}\text{be the diameter and}\mathit{\text{A}}\text{be the area of the circle at any time}\mathit{\text{t}}\text{. Then,} \\ A=\mathrm{\pi }{r}^2\left(\mathrm{where}\mathit{}r\mathrm{is}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{cicle}\right) \\ \Rightarrow \mathit{\text{A}}\text{=π}\frac{D^2}{4}\left[\because r=\frac{D}{2}\right] \\ \Rightarrow \frac{dA}{dt}=2\mathrm{\pi }\frac{D}{4}\frac{dD}{dt} \\ \Rightarrow \frac{dA}{dt}=\frac{\mathrm{\pi }}{2}\times 2\mathrm{\pi }\times 1\left[\because \frac{dD}{dt}=1\mathrm{cm}/\sec \right] \\ \Rightarrow \frac{dA}{dt}={\mathrm{\pi }}^2{\mathrm{cm}}^2/\sec \end{gathered}$$