Gas leaks out of a spherical balloon at 50 cm³/s. How fast is the radius of the balloon decreasing when the radius is 5 cm?

Explanation

1. 01

   $V = \frac{4}{3}\pi r^{3}$, so $\frac{dV}{dt} = 4\pi r^{2}\frac{dr}{dt}$.

2. 02

   $-50 = 4\pi \times 25 \times \frac{dr}{dt}$, so $\frac{dr}{dt} = -\frac{50}{100\pi} = -\frac{1}{2\pi}$: the radius falls at $\frac{1}{2\pi}$ cm/s.