Problem: For time 0 ≤ t ≤ 10, water is flowing into a small tub at a rate given by the function F defined by F(t) = arctan (π/2 - t/10). For time 5 ≤ t ≤ 10, water is leaking from the tub at a rate given by the function L defined by L(t) = 0.03 (20t - t2 - 75). Both F(t) and L(t) are measured in cubic feet per minute, and t is measured in minutes. The volume of water in the tub, in cubic feet, at time t minutes is given by W(t).
(a) At time t = 3, there are 2.5 cubic feet of water in the tub. Write an equation for the locally linear approximation of Watt = 3, and use it to approximate the volume of water in the tub at time t = 3.5.
(b) Find W" (8). Using correct units, interpret the meaning of W" (8) in the context of the problem.
(c) Is there a time t, for 5 < t < 10, at which the rate of change of the volume of water in the tub changes from positive to negative? Give a reason for your answer.
(d) The tub is in the shape of a rectangular box that is 0.5 foot wide, 4 feet long, and 3 feet deep. What is the rate of change of the depth of the water in the tub at time t = 6?