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

Options

• \[\frac{2}{\pi} \text { unit }\]

•  \[\frac{1}{\pi} \text { unit }\]

• \[\frac{\pi}{2} \text { units }\]

• π units

Answer 1

 \[\frac{1}{\pi} \text { unit }\]

\[\text { Let r be the radius andAbe the area of the circle at any time t. Then,}\]

\[A=\pi r^2 \]

\[ \Rightarrow A = \frac{\pi D^2}{4} \left[ \because r = \frac{D}{2} \right]\]

\[ \Rightarrow \frac{dA}{dt} = \frac{\pi D}{2}\frac{dD}{dt}\]

\[ \Rightarrow \frac{dD}{dt} = \frac{\pi D}{2}\frac{dD}{dt} \left[ \because \frac{dA}{dt} = \frac{dD}{dt} \right]\]

\[ \Rightarrow \frac{D}{2} = \frac{1}{\pi}\]

\[ \Rightarrow r = \frac{1}{\pi} \text { units }\]