Question

Gas is being pumped into a spherical balloon. Then, the rate at of $30 \, \text{ft}^3/\text{min}$ which the radius increases when it reaches the value 15 ft, is

• A. $\frac{1}{5} \, \frac{\text{ft}}{\text{min}}$
• B. $\frac{1}{15} \, \frac{\text{ft}}{\text{min}}$
• C. $\frac{1}{20} \, \frac{\text{ft}}{\text{min}}$
• D. $\frac{1}{25} \, \frac{\text{ft}}{\text{min}}$

Hint:

To find the rate at which the radius changes in spherical volume problems, differentiate the volume formula and substitute the known rate of change of volume.

Answer 1

The Correct Option is A

We are given that the rate at which the volume of the balloon changes is: \[ \frac{dV}{dt} = 30 \, \text{ft}^3/\text{min} \] The volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] To find the rate at which the radius changes, differentiate the volume with respect to time: \[ \frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt} \] Substitute the values: \[ 30 = 4 \pi (15)^2 \frac{dr}{dt} \] Solve for $\frac{dr}{dt}$: \[ \frac{dr}{dt} = \frac{30}{4 \pi (15)^2} = \frac{1}{5} \, \frac{\text{ft}}{\text{min}} \]