MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
During a 12-hour manufacturing shift, the volume of water in a storage tank is modeled by a continuous function V(t), measured in gallons, where t is the number of hours after the shift begins. Volume readings at four times are recorded as follows:
t (h) 0 3 6 9 12
V(t) (gal) 250 420 180 500 250
(a) Demonstrate that there are at least two distinct times during the shift when the tank contained exactly 300 gallons. Specify intervals that each contain such a time and justify thoroughly.
(b) Explain why the tank must have contained every volume between 180 gallons and 500 gallons at some time during the shift. Support your argument with appropriate reasoning.
(c) The tank has a maximum capacity of 550 gallons. Based on the given information, discuss whether it is certain, possible, or impossible that the tank overflowed at some point during the shift. Provide a clear justification.