Question

The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing in mm³/s when the diameter is 60 mm? Round your answer to two decimal places.

Answer 1

The volume of a sphere with a diameter of 60 mm is increasing at a rate of approximately 45238.93 mm³/s when the radius is increasing at 4 mm/s.

Step-by-step explanation:

To determine how fast the volume of a sphere is increasing when its diameter is 60 mm, we use the rate of change of volume with respect to time, given by the formula dV/dt = 4πr^2 (dr/dt). Here, dV/dt is the rate of change of volume, r is the radius, and dr/dt is the rate of change of the radius. Given that the radius is increasing at a rate of 4 mm/s (dr/dt = 4 mm/s), and the radius r is half of the diameter, so r = 60 mm / 2 = 30 mm. Substituting the values, we get:

dV/dt = 4π(30^2)(4)

Computing this gives us dV/dt = 4π(900)(4) = 14400π mm³/s.

Therefore, the volume is increasing at a rate of approximately 45238.93 mm³/s when rounded to two decimal places.