Consider a cylindrical water tank of constant cross

Chapter 2, Problem 19

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Consider a cylindrical water tank of constant cross section A. Water is pumped into thetank at a constant rate k and leaks out through a small hole of area a in the bottomof the tank. From Torricellis principle in hydrodynamics (see in Section 2.3)it follows that the rate at which water flows through the hole is a2gh, where h is thecurrent depth of water in the tank,g is the acceleration due to gravity, and is a contractioncoefficient that satisfies 0.5 1.0.(a) Show that the depth of water in the tank at any time satisfies the equationdh/dt = (k a2gh )/A.(b) Determine the equilibrium depth he of water, and show that it is asymptotically stable.Observe that he does not depend on A.Harvesting a Renewable Resource. Suppose that the population y of a certain species of fish(for example, tuna or halibut) in a given area of the ocean is described by the logistic equationdy/dt = r(1 y/K)y.Although it is desirable to utilize this source of food, it is intuitively clear that if too many fishare caught, then the fish population may be reduced below a useful level and possibly evendriven to extinction. 20 and 21 explore some of the questions involved in formulatinga rational strategy for managing the fishery.15