Question

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

• A. $8\pi {cm}^2/s$
• B. $12\pi {cm}^2/s$
• C. $160\pi {cm}^2/s$
• D. $200\pi {cm}^2/s$

Answer 1

The correct option is C $160\pi {cm}^2/s$

Let $r$ be the radius and $S$ be the surface area of the sphere at any time $t$.
Then, $S=4\pi r^2$ .....(i)
Given, $r=200cm$ and $\frac{dr}{dt}=0.1cm/s$
Now differentiating equation (i) with respect to $t$, we get
$\frac{dS}{dt}=4\pi \cdot 2r\frac{dr}{dt}$
$\Rightarrow {\left(\frac{dS}{dt}\right)}_{r=200}=8\pi \times 200\times 0.1$
$=160\pi {cm}^2/s$