Question

The rate of change of the surface area of the sphere of radius $r$ when the radius is increasing at the rate of $2$ cm/sec is proportional to :

• A. $\frac{1}{r^2}$
• B. $\frac{1}{r}$
• C. $r^2$
• D. $r$

Answer 1

The correct option is D $r$

The surface area of a sphere is given by $4\pi r^2,r$ being the radius.

Also, $\frac{dr}{dt}=2$ cm/sec

$\frac{d(S.A.)}{dt}=\frac{d\left(4\pi r^2\right)}{dt}=4\pi \times 2r\frac{dr}{dt}=4\pi \times 2r\times 2=16\pi r$

Therefore, it is proportional to $r$.