Question

A spherical balloon is expanding. If the radius is increasing at the rate of 2 centimeters per minute, the rate at which the volume increases (in cubic centimeters per minute) when the radius is 5 centimeters is

• A. \(10\pi\)
• B. \(100\pi\)
• C. \(200\pi\)
• D. \(50\pi\)

Hint:

For sphere: \(\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\). Always substitute radius at the instant asked.

Answer 1

The Correct Option is C

Step 1: Write volume of sphere.
\[ V=\frac{4}{3}\pi r^3 \]
Step 2: Differentiate w.r.t. time \(t\).
\[ \frac{dV}{dt}=\frac{d}{dt}\left(\frac{4}{3}\pi r^3\right) =4\pi r^2 \frac{dr}{dt} \]
Step 3: Substitute given values.
\[ r=5,\quad \frac{dr}{dt}=2 \]
\[ \frac{dV}{dt}=4\pi (5)^2(2) =4\pi \cdot 25 \cdot 2 =200\pi \]
Final Answer:
\[ \boxed{200\pi} \]