Water flows into and out of a storage tank at a net rate modeled (in gallons per…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Water flows into and out of a storage tank at a net rate modeled (in gallons per hour) by

f(t)=10+8\sin\!\left(\frac{\pi t}{6}\right)-4t, \qquad 0\le t\le6,

where

t

is the elapsed time in hours after midnight. At midnight the tank contains 150 gallons of water. Define

V(x)=150+\int_{0}^{x} f(t)\,dt, \qquad 0\le x\le6,

so that

V(x)

gives the volume of water (in gallons) in the tank at time

t=x

. (a) Determine the time intervals on which the volume of water in the tank is increasing and the intervals on which it is decreasing during the six-hour period. Show analytical work that supports your conclusions. (b) Find the exact time

t

in the interval

(0,6)

at which the tank first reaches its greatest volume. Justify that this time gives the absolute maximum of

V

[0,6]

. (c) Calculate the greatest volume of water present in the tank during the six-hour period. Show all steps in your computation. (d) Interpret the meaning of the value

V(6)

in the context of the problem, and indicate whether the tank contains more or less water at 6 a.m. than it did at midnight. Support your answer with appropriate reasoning.

Related theory

Browse the problem library