A spherical ice ball is melting at the rate of 100π cm3/min. The rate…

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

Let r be the radius and V the volume of the ball at time t:

    \(\displaystyle V = \frac{4}{3}\pi r^3\)

Differentiating with respect to t by the chain rule:

    \(\displaystyle \frac{dV}{dt} = 4\pi r^2\,\frac{dr}{dt}\)

The volume decreases at \(100\pi\) cm³/min, so \(\frac{dV}{dt} = -100\pi\). At r = 15 cm:

    \(\displaystyle -100\pi = 4\pi(15)^2\,\frac{dr}{dt} = 900\pi\,\frac{dr}{dt}\)

    \(\displaystyle \frac{dr}{dt} = -\frac{100\pi}{900\pi} = -\frac{1}{9}\)

The minus sign shows that the radius is decreasing, at the rate of \(\frac{1}{9}\) cm/min.