Consider a cylindrical water tank of constant cross section A. Water is pumped into the

Chapter 2, Problem 19

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Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. FromTorricellis principle in hydrodynamics (see in Section 2.3) it follows that the rate at which water flows through the hole is a 2gh, where h is the current depth of water in the tank, g is the acceleration due to gravity, and is a contraction coefficient that satisfies 0.5 1.0. (a) Show that the depth of water in the tank at any time satisfies the equation dh/dt = (k a 2gh )/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 equation dy/dt = r(1 y/K)y.