The radius of a sphere is changing at the rate of 0.1 cm/sec.
Question:

The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is

(a) $8 \pi \mathrm{cm}^{2} / \mathrm{sec}$

(b) $12 \pi \mathrm{cm}^{2} / \mathrm{sec}$

(c) $160 \pi \mathrm{cm}^{2} / \mathrm{sec}$

(d) $200 \mathrm{~cm}^{2} / \mathrm{sec}$

Solution:

(c) $160 \pi \mathrm{cm}^{2} / \mathrm{sec}$

Let $r$ be the radius and $S$ be the surface area of the sphere at any time $t .$ Then,

$S=4 \pi r^{2}$

$\Rightarrow \frac{d S}{d t}=8 \pi r \frac{d r}{d t}$

$\Rightarrow \frac{d S}{d t}=8 \pi(200)(0.1)$

$\Rightarrow \frac{d S}{d t}=160 \pi \mathrm{cm}^{2} / \mathrm{sec}$