Question

A spherical balloon is filled with $4500\,\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\,\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is

• A. $\frac{9}{7}$
• B. $\frac{7}{9}$
• C. $\frac{2}{9}$
• D. $\frac{9}{2}$

Answer 1

Answer: (C)

$\frac{2}{9}$

Answer 2

$V = \frac{4}{3}\pi r^{3}\quad\quad4500 \pi = \frac{4\pi r^{3}}{3}$ $\frac{dV}{dt} = 4\pi r^{2} \left(\frac{dr}{dt}\right)\quad\quad45 ??25 ??3 = r^{3}$ $r = 15\, m$ after $49$ min $= \left(4500 - 49.72\right)\pi = 972\, \pi\,m^{3}$ $972\,\pi = \frac{4}{3} \pi r^{3}$ $r^{3} = 3??43 = 3??3^{5}$ $r = 9$ $72\, \pi = 4\pi ??9 ??9 \left(\frac{dr}{dt}\right)$ $\frac{dr}{dt} = \left(\frac{2}{9}\right)$