Question

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to
(a) $\frac{2}{\mathrm{\pi }}\mathrm{unit}$

(b) $\frac{1}{\mathrm{\pi }}\mathrm{unit}$

(c) $\frac{\mathrm{\pi }}{2}\mathrm{units}$

(d) π units

Answer 1

(b) $\frac{1}{\mathrm{\pi }}\mathrm{unit}$

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}A\text{be the area of the circle at any time}\mathit{\text{t}}\text{. Then,} \\ \mathit{\text{A}}\text{=}\mathrm{\pi }{r}^2\mathit{} \\ \mathit{\Rightarrow }A=\frac{\pi D^{\mathit{2}}}{4}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\left[\mathit{\because }\mathit{}\mathit{}r\mathit{=}\frac{D}{\mathit{2}}\right] \\ \Rightarrow \frac{dA}{dt}=\frac{\pi D}{2}\frac{dD}{dt} \\ \Rightarrow \frac{dD}{dt}=\frac{\mathrm{\pi }D}{2}\frac{dD}{dt}\left[\because \frac{dA}{dt}=\frac{dD}{dt}\right] \\ \Rightarrow \frac{D}{2}=\frac{1}{\mathrm{\pi }} \\ \Rightarrow r=\frac{1}{\mathrm{\pi }}\mathrm{units} \end{gathered}$$