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−t^2−75) . 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 W at t=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 ?
