please help with this

Question

A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 9 cm. (Note the answer is a positive number).

Answer 1

V = 4/3 π r^3, r = D/2 so V = 4/3 π (D/2)^3 = 4/3 π D^3/8 or

V = 1/6 π D^3

Deriving with respect to time using the chain rule we

dV/dt = 1/2 π D^2 dD/dt

They gave us D = 9 cm and dD/dt = -0.4 cm/min. Plugging these in we get

dV/dt = 1/2 π (9 cm)^2 * (-0.4 cm/min) = -1/2 π * 81 cm² * 0.4 cm/min = -16.2 π or

decreasing at approximately 50.9 cm³/min

Hope this helps! Let me know if I can help with any of the steps. Thank you!