Question

If the radius of a sphere is increasing at a constant rate of 1 cm/sec, then at what rate is the volume of the sphere increasing?

1) 4π cm³/sec
2) 2π cm³/sec
3) π cm³/sec
4) 3π cm³/sec

Answer 1

The rate at which the volume of the sphere is increasing is 4π cm³/sec.

Step-by-step explanation:

To find the rate at which the volume of a sphere is increasing, we can use the formula for the volume of a sphere: V = (4/3)πr³, where r is the radius. Since the radius is increasing at a constant rate, we can differentiate the formula with respect to time (t):

dV/dt = (4/3)π(3r²)(dr/dt)

Substituting the given values, we have:

dV/dt = (4/3)π(3r²)(1)

dV/dt = 4πr²

Therefore, the rate at which the volume of the sphere is increasing is 4πr² cm³/sec. Since the radius is increasing at a rate of 1 cm/sec, the rate of increase in volume is 4π(1)² = 4π cm³/sec.