Question

A spherical balloon is filled with (4500\pi) cubic meters of helium gas. If a leak causes gas to escape at (72\pi) m(^3)/min, then the rate at which the radius decreases 49 minutes after leakage began is

• A. (\frac{9}{7})
• B. (-\frac{2}{9})
• C. (\frac{9}{2})
• D. [suspicious link removed]

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)