A cylindrical holding tank contains a saline solution whose volume,
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A cylindrical holding tank contains a saline solution whose volume, V, is recorded every 1 minute as the solution is pumped out. The table below lists the recorded volume (in liters) at selected times t (in minutes), where t=0 corresponds to the moment pumping begins.
Time t (min): 0 1 2 3 4 5 6 7 8
Volume V (L): 420 409 392 370 343 310 274 235 193
(a) Calculate the average rate at which the volume is changing on the interval [2,7]. Include proper units.
(b) Describe what the value found in part (a) means in the context of the pumping process.
(c) Compute, to the nearest liter per minute, the average rate of change on each one-minute sub-interval from t=2 to t=7. Based on these values, explain whether the instantaneous rate of volume change appears to be increasing, decreasing, or remaining roughly constant over that time period.
(d) Suppose engineers wish to estimate the instantaneous pumping rate at t=5. Using only the data provided, state one numerical estimate for this instantaneous rate and justify why your estimate is reasonable.