The radius of a circle is increasing at the uniform rate of $3 \text{ cm/s}$. At the instant the radius of the circle is $2 \text{ cm}$, then at what rate area increases?

$12\pi \text{ cm}^2/\text{s}$

The correct answer is Option (3) → $12\pi \text{ cm}^2/\text{s}$ ##

Radius of the circle, $r = 2 \text{ cm}$.

Rate of change of the radius with respect to time, $\frac{dr}{dt} = 3 \text{ cm/s}$.

$ A = \pi r^2 $

$ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} $

$ \left( \frac{dA}{dt} \right)_{r=2} = 2\pi(2)(3) = 12\pi \text{ cm}^2/\text{sec} $