A spherical ice ball is melting at the rate of $100\pi cm^3/min$. The rate at which its radius is decreasing when its radius is 15 cm, is

$\frac{1}{9}\text{cm/min}$

The correct answer is Option (2) → $\frac{1}{9}\text{cm/min}$

Let the radius of the spherical ice ball be \( r \), and its volume be

$ V = \frac{4}{3} \pi r^3 $

Differentiate both sides with respect to time \( t \):

$ \frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt} $

Given:

$ \frac{dV}{dt} = -100\pi \ \text{cm}^3/\text{min} $ (negative because the volume is decreasing)

$ r = 15\ \text{cm} $

Substitute into the differentiated equation:

$ -100\pi = 4\pi (15)^2 \cdot \frac{dr}{dt} $

$ -100\pi = 4\pi \cdot 225 \cdot \frac{dr}{dt} $

$ -100\pi = 900\pi \cdot \frac{dr}{dt} $

Divide both sides by \( 900\pi \):

$ \frac{dr}{dt} = \frac{-100\pi}{900\pi} = -\frac{1}{9} $