A pond forms as water collects in a conical depression of radius a and depth h. Suppose

Chapter 2, Problem 18

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A pond forms as water collects in a conical depression of radius a and depth h. Suppose that water flows in at a constant rate k and is lost through evaporation at a rate proportional to the surface area. a. Show that the volume V(t) of water in the pond at time t satisfies the differential equation dV dt = k (3a/ h)2/3V2/3, where is the coefficient of evaporation. b. Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable? c. Find a condition that must be satisfied if the pond is not to overflow. 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 = r1 y K _y. Although it is desirable to utilize this source of food, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level and possibly even driven to extinction. 19 and 20 explore some of the questions involved in formulating a rational strategy for managing the fishery.16