A storage tank is continuously monitored for its volume of liquid (in liters) over a…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A storage tank is continuously monitored for its volume of liquid V (in liters) over a 24-hour period. A technician summarizes the graph of V as follows:
• From t=0 to t=6 hours, V increases from 460 L to 500 L with decreasing slope.
• Between t=6 and t=14 hours, V decreases to 440 L; the steepest decrease occurs near t=10 h.
• From t=14 to t=18 hours, V rises to 530 L; the slope is positive throughout but decreases toward t=18 h.
• From t=18 to t=24 hours, V decreases, ending the day at 510 L.
Assume V is differentiable for 0≤t≤24 and that the description captures all changes in monotonicity and concavity.
(a) Identify all values of t in 0≤t≤24 at which V could have a local extremum. Give a brief reason for each.
(b) Determine the time at which V attains its absolute maximum on [0,24]. Justify your answer based on the information provided.
(c) Provide an estimate of this maximum volume.
(d) Explain, in practical terms, why knowing this maximum is important for managing the storage tank.