Water is pumped into and out of a storage tank throughout the day
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Water is pumped into and out of a storage tank throughout the day. The net rate of change of the volume of water in the tank is modeled by a continuous function r (in liters per hour) for 0≤t≤20, where t is the time in hours after midnight. A verbal description of r is given:
• r(t) is positive and increases linearly from 0 at t=0 to a maximum of 30 at t=4.
• From t=4 to t=12, r(t) decreases linearly, reaching 0 at t=12.
• From t=12 to t=16, r(t) is constant at −15.
• From t=16 to t=20, r(t) increases linearly from −15 to 0.
At midnight (t=0) the tank contains 450 liters of water.
Respond fully to each part.
(a) Determine the volume of water in the tank at t=12.
(b) Find the total amount of water that has entered the tank during the interval 0≤t≤20.
(c) At what time is the volume of water in the tank minimal? Justify your answer.
(d) Write a piecewise algebraic expression for a differentiable function V representing the volume of water in the tank for 0≤t≤20 and verify that V′(t)=r(t).