The volume of a spherical hot air balloon V(r) =(4/3)pir^3 changes as its radius changes

Question

The volume of a spherical hot air balloon V(r) =(4/3)pir^3 changes as its radius changes. The radius is a function of time given by r(t) = 3t.

 Find the average rate of change of the volume with respect to t as t changes from t = 1

 to t = 4.

Answer 1

The average rate of change of volume is (V(t₂)-V(t₁))/(t₂-t₁)

V = (4/3) πr³ = (4/3)π(3t)³ = 36πt³

Now just plug into the formulas with t₂=4 and t₁=1

Please consider a tutor. Take care.

You could have calculated the integral from 1 to 4 of (dV/dt) dt divided by (4-1), but there is no need to derive dV/dt from related rates as the integral is V(t).