Hot Tub Problem: Figure 7-3k shows a cylindrical hot tub 8 ft in diameter and 4 ft deep

Chapter 7, Problem 5

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Hot Tub Problem: Figure 7-3k shows a cylindrical hot tub 8 ft in diameter and 4 ft deep. At time t = 0 min, the drain is opened and water flows out. The rate at which it flows is proportional to the square root of the depth, y, in feet. Because the tub has vertical sides, the rate is also proportional to the square root of the volume, V, in cubic feet, of water left. Figure 7-3k a. Write a differential equation for the rate at which water flows from the tub. That is, write an equation for dV/dt in terms of V . b. Separate the variables and integrate the differential equation you wrote in part a. Transform the result so that V is expressed explicitly in terms of t. Tell how V varies with t. c. Suppose that the tub initially (when t = 0) contains 196 ft3 of water and that when thedrain is first opened the water flows out at28 ft3/min (when t = 0, dV/dt = 28). Findthe particular solution of the differentialequation that fits these initial conditions.d. Naive thinking suggests that the tub will beempty after 7 min since it contains 196 ft3and the water is flowing out at 28 ft3/min.Show that this conclusion is false, andjustify your answer e. Does this mathematical model predict atime when the tub is completely empty, ordoes the volume, V, approach zeroasymptotically? If there is a time, state it.f. Draw a graph of V versus t in a suitabledomain.g. See in Section 7-7 to see whathappens if a hose is left running in thehot tub while it drains.