Water drains from a cylindrical tank through an outlet at the bottom
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Water drains from a cylindrical tank through an outlet at the bottom. Let V(t) denote the volume of water (in liters) in the tank at time t minutes for 0≤t≤25. The function V is continuous on [0,25] and differentiable on (0,25). At t=0 the tank holds 150 liters, and at t=25 the tank holds 40 liters.
(a) Describe, in complete sentences, why the given properties of V ensure that there is at least one time c in (0,25) at which V′(c) equals the average rate at which water leaves the tank during the 25-minute interval.
(b) Without performing any calculations, explain how you could determine whether there must also be a time at which the instantaneous outflow rate ∣V′(t)∣ exceeds the average outflow rate. Support your explanation with appropriate reasoning.