The volume of a spherical hot air balloon v(r) = 4/3 πr^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.

At t = 1: The radius is 3(1) = 3 The volume is 4/3 π(3)^3 = 113 At t = 4: The radius is 3(4) = 12 The volume is 4/3 π(12)^3 = 7238 We can find the change in the volume: The change in volume is 7238 - 113 = 7125 The change in volume occurred during a time of 3 seconds. To find the average rate of change in the volume, we need to divide the change in the volume by the time. average rate of change = 7125 / 3 = 2375 The average rate of the change in the volume is 2375 (The units should be (m^3/s) or (ft^3/s) or (cm^3/s) etc...)