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 has begun, is

• A. \(\frac{9}{7}\)
• B. \(-\frac{2}{9}\)
• C. \(\frac{9}{2}\)
• D. \(\frac{2}{9}\)

Hint:

Always calculate the specific value of the independent variable (like radius) at the requested time first.

Answer 1

The Correct Option is D

Step 1: Volume at (t = 49)
Initial (V = 4500\pi). Gas escaped = (72\pi \times 49 = 3528\pi).
Current (V = 4500\pi - 3528\pi = 972\pi).

Step 2: Find current radius
(\frac{4}{3}\pi r^3 = 972\pi \implies r^3 = 729 \implies r = 9).

Step 3: Use rate of change
(V = \frac{4}{3}\pi r^3 \implies \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}).
(-72\pi = 4\pi (9)^2 \frac{dr}{dt}).
(\frac{dr}{dt} = \frac{-72}{324} = -\frac{2}{9}).

Step 4: Magnitude
The radius decreases at a rate of (\frac{2}{9}) m/min.
Final Answer: (D)