If the rate of change of area of a circle is equal

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

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