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) $\frac{2}{\pi}$ unit
(b) $\frac{1}{\pi}$ unit
(c) $\frac{\pi}{2}$ units
(d) $\pi$ units
Solution:
(b) $\frac{1}{\pi}$ unit
Let $r$ be the radius and $A$ be the area of the circle at any time $t$. Then,
$A=\pi r^{2}$
$\Rightarrow A=\frac{\pi D^{2}}{4}$ $\left[\because r=\frac{D}{2}\right]$
$\Rightarrow \frac{d A}{d t}=\frac{\pi D}{2} \frac{d D}{d t}$
$\Rightarrow \frac{d D}{d t}=\frac{\pi D}{2} \frac{d D}{d t}$ $\left[\because \frac{d A}{d t}=\frac{d D}{d t}\right]$
$\Rightarrow \frac{D}{2}=\frac{1}{\pi}$
$\Rightarrow r=\frac{1}{\pi}$ units