Let the rate of change of water (in liters per minute) flowing into and out of a tank…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let the rate of change of water (in liters per minute) flowing into and out of a tank be modeled by the piecewise function
r(t)=⎩⎨⎧6t,4,−2(t−3),0≤t<1,1≤t<3,3≤t≤4,
where t is measured in minutes. At time t=0 the tank contains 20 liters of water.
(a) Determine the exact time(s) in 0≤t≤4 at which the flow rate changes sign.
(b) Write, but do not evaluate, a sum of definite integrals whose value equals the net change in the amount of water in the tank from t=0 to t=4. Be sure to split the integrals at every time you listed in part (a) and at any other points required by the definition of r(t).
(c) Evaluate the integral expression from part (b) to find the volume of water in the tank at t=4 minutes.
(d) Compute the total volume of water that passed through the pipe (in either direction) during the interval 0≤t≤4. Explain how this quantity differs conceptually from the net change found in part (c).