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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
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Solution
\[\frac{1}{\pi} \text { unit }\]
\[\text { Let r be the radius andAbe the area of the circle at any time t. Then,}\]
\[A=\pi r^2 \]
\[ \Rightarrow A = \frac{\pi D^2}{4} \left[ \because r = \frac{D}{2} \right]\]
\[ \Rightarrow \frac{dA}{dt} = \frac{\pi D}{2}\frac{dD}{dt}\]
\[ \Rightarrow \frac{dD}{dt} = \frac{\pi D}{2}\frac{dD}{dt} \left[ \because \frac{dA}{dt} = \frac{dD}{dt} \right]\]
\[ \Rightarrow \frac{D}{2} = \frac{1}{\pi}\]
\[ \Rightarrow r = \frac{1}{\pi} \text { units }\]
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