Question

The radius of a sphere initially at zero increases at the rate of 5 cm/sec. Then its volume after 1 sec is increasing at the rate of :

• A. $50\pi$
• B. $5\pi$
• C. $500\pi$
• D. none of these

Answer 1

Answer: (C)

Let ‘r’ be the radius and V be the volume of the sphere.
Given : Radius increases at the rate of 5cm/sec.
$\therefore \frac{dr}{dt}=5cm/\sec$
Now, $V=\frac{4}{3}\pi r^3$
$\therefore \frac{dV}{dt}=\frac{4}{3}\pi (3r^2)\frac{dr}{dt}=4\pi r^2(5)=20\pi r^2$
Now, after one second, r = 5
$\therefore \frac{dV}{dt}$ after $1\sec =20\pi (5{)}^2=500\pi$