Question

The radius r of a spherical balloon is expanding at the constant rate of 14 in/min. (a) Determine the rate at which the volume V changes with respect to time, in in3 /min, when r = 8 in. Round your answer to the nearest integer. Recall V = 4 3 πr3 .

Answer 1

The rate at which the volume of the balloon changes with respect to time is 5635 in³/min when the radius is 8 in.

Step-by-step explanation:

To determine the rate at which the volume V changes with respect to time, we need to find dV/dt, which represents the derivative of the volume function with respect to time. Given the formula V = 4/3 πr³ for the volume of a sphere, we can substitute the given rate of change of the radius, dr/dt = 14 in/min, into the formula and differentiate both sides with respect to time.

dV/dt = d/dt(4/3 πr³)

= 4πr² (dr/dt)

= 4π(8)² (14)

= 1792π in³/min

Therefore, when r = 8 in, the rate at which the volume V changes with respect to time is approximately 1792π in³/min. Rounding this to the nearest integer, the volume is changing at a rate of 5635 in³/min.