Chapter 13

Assume that the environmental carrying capacity \(N_{u}\) is determined principally by the availability of food. Argue that under such an assumption, as \(N\) approaches \(N_{u}\) the physical condition of the average fish deteriorates as competition for the food supply becomes more severe. What does this suggest about the survival of the species when natural disasters such as storms, severe winters, and similar circumstances further restrict the food supply? Where should conservationists attempt to maintain the population level?

Find the local maximum value of the function $$ f(x, y)=x y-x^{2}-y^{2}-2 x-2 y+4 $$

In 1981 and 1982 , the deer population in the Florida Everglades was very high. Although the deer were plentiful, they were on the brink of starvation. Hunting permits were issued to thin out the herd. This action caused much furor on the part of environmentalists and conservationists. Explain the poor health of the deer and the purpose of the special hunting permits in terms of population growth and population submodels.

Find the local minimum value of the function $$ f(x, y)=3 x^{2}+6 x y+7 y^{2}-2 x+4 y $$

Use the method of Lagrange multipliers to solve Problems. Find the minimum distance from the surface \(x^{2}+y^{2}-z^{2}=1\) to the origin.

A differentiable function \(f(x, y)\) has a saddle point at a point \((a, b)\) where its partial derivatives are simultaneously zero, if in every open disk centered at \((a, b)\) there are domain points where \(f(x, y)>f(a, b)\) and domain points where \(f(x, y)

Use the method of Lagrange multipliers to solve Problems. Find three numbers whose sum is 9 and whose sum of squares is as small as possible.

A continuous function \(f(x, y)\) takes on its absolute extrema on a closed and bounded region either at an interior point or at a boundary point of the region. Find the absolute extrema of $$ f(x, y)=48 x y-32 x^{3}-24 y^{2} $$ on the square region \(0 \leq x \leq 1\) and \(0 \leq y \leq 1\).

One of the key assumptions underlying the models developed in this section is that the harvest rate equals the growth rate for a sustainable yield. The reproduction submodels in Figures \(13.19\) and \(13.22\) suggest that if the current population levels are known, it is possible to estimate the growth rate. The implication of this knowledge is that if a quota for the season is established based on the estimated growth rate, then the fish population can be maintained, increased, or decreased as desired. This quota system might be implemented by requiring all commercial fishermen to register their catch daily and then closing the season when the quota is reached. Discuss the difficulties in determining reproduction models precise enough to be used in this manner. How would you estimate the population level? What are the disadvantages of having a quota that varies from year to year? Discuss the practical political difficulties of implementing such a procedure.

A company manufactures \(x\) floor lamps and \(y\) table lamps each day. The profit in dollars for the manufacture and sale of these lamps is $$ P(x, y)=18 x+2 y-0.05 x^{2}-0.03 y^{2}+0.02 x y-100 $$ Find the daily production level of each lamp to maximize the company's profits.

Problems 5 and 6 are related to fishing regulation. One of the key assumptions underlying the models developed in this section is that the harvest rate equals the growth rate for a sustainable yield. The reproduction submodels in Figures \(13.19\) and \(13.22\) suggest that if the current population levels are known, it is possible to estimate the growth rate. The implication of this knowledge is that if a quota for the season is established based on the estimated growth rate, then the fish population can be maintained, increased, or decreased as desired. This quota system might be implemented by requiring all commercial fishermen to register their catch daily and then closing the season when the quota is reached. Discuss the difficulties in determining reproduction models precise enough to be used in this manner. How would you estimate the population level? What are the disadvantages of having a quota that varies from year to year? Discuss the practical political difficulties of implementing such a procedure.

Use the method of Lagrange multipliers to solve Problems 3-6. Find the hottest point \((x, y, z)\) along the elliptical orbit $$ 4 x^{2}+y^{2}+4 z^{2}=16 $$ where the temperature function is $$ T(x, y, z)=8 x^{2}+4 y z-16 z+600 $$

One of the difficulties in managing a fishery in a free-enterprise system is that excess capacity may be created through overcapitalization. This happened in 1970 when the capacity of the Peruvian anchoveta fishermen was sufficient to catch and process the maximum annual growth rate in less than 3 months. A disadvantage of restricting access to the fishery by closing the season after a quota is reached is that this excess capacity is idle during much of the season, which creates a politically and economically unsatisfying situation. An alternative is to control the capacity in some manner. Suggest several procedures for controlling the capacity that is developed. What difficulties would be involved in implementing a procedure such as restricting the number of commercial fishing permits issued?

If \(x\) and \(y\) are the amounts of labor and capital, respectively, to produce $$ Q(x, y)=0.54 x^{2}-0.02 x^{3}+1.89 y^{2}-0.09 y^{3} $$ units of output for manufacturing a product, find the values of \(x\) and \(y\) to maximize \(Q\).

Consider an athlete competing in the shot put. What factors influence the length of his or her throw? Construct a model that predicts the distance thrown as a function of the initial velocity and angle of release. What is the optimal angle of release? If the athlete cannot maximize the initial velocity at the angle of release you propose, should he or she be more concerned with satisfying the angle of release or generating a high initial velocity? What are the trade-offs?

Problems 5 and 6 are related to fishing regulation. One of the difficulties in managing a fishery in a free-enterprise system is that excess capacity may be created through overcapitalization. This happened in 1970 when the capacity of the Peruvian anchoveta fishermen was sufficient to catch and process the maximum annual growth rate in less than 3 months. A disadvantage of restricting access to the fishery by closing the season after a quota is reached is that this excess capacity is idle during much of the season, which creates a politically and economically unsatisfying situation. An alternative is to control the capacity in some manner. Suggest several procedures for controlling the capacity that is developed. What difficulties would be involved in implementing a procedure such as restricting the number of commercial fishing permits issued?

Use the method of Lagrange multipliers to solve Problems 3-6. A Least Squares Plane-Given the four points \(\left(x_{k}, y_{k}, z_{k}\right)\) $$ (0,0,0),(0,1,1),(1,1,1),(1,0,-1) $$ find the values of \(A, B\), and \(C\) to minimize the sum of squared errors $$ \sum_{k=1}^{4}\left(A x_{k}+B y_{k}+C-z_{k}\right)^{2} $$ if the points must lie in the plane $$ z=A x+B y+C $$

The total cost to manufacture one unit of product \(A\) is \(\$ 3\), and for one unit of product \(B\) it is \(\$ 2\). If \(x\) and \(y\) are the retail prices per unit of \(A\) and \(B\), respectively, then marketing research has established that $$ Q_{A}=2750-700 x+200 y $$ and $$ Q_{B}=2400+150 x-800 y $$ are the quantities of each product that will be sold each day. Find a function \(P(x, y)\) modeling the daily profit and the maximum daily profit.

John Smith is responsible for periodically buying new trucks to replace older trucks in his company's fleet of vehicles. He is expected to determine the time a truck should be retained so as to minimize the average cost of owning the truck. Assume the purchase price of a new truck is \(\$ 9000\) with trade- in. Also assume the maintenance cost (in dollars) per truck for \(t\) years can be expressed analytically by the following empirical model: $$ C(t)=640+180(t+1) t $$ where \(t\) is the time in years that the company owns the truck. a. Determine \(E(t)\), the total cost function for a single truck retained for a period of \(t\) years. b. Determine \(E_{A}(t)\), the average annual cost function for a single truck that is kept in the fleet for \(t\) years. c. Graphically depict \(E_{A}(t)\) as a function of \(t\). Justify the shape of your graph. d. Analytically determine \(t^{*}\), the optimal period that a truck should be retained in the fleet. Remember that the objective is to minimize the average cost of owning a truck. e. Suppose we have to round \(t^{*}\) to the nearest whole year. In general, would it be better to round up or round down? Justify your answer.

An electric power-generating company charges different rates for residential and business users. (You might consider some reasons why this would be so.) The cost of producing the electricity is the same for all users and equals \(\$ 1000\) in fixed costs plus an additional \(\$ 200\) for each unit produced. If residential customers use \(x\) units of electricity, they pay \(p=1200-2 x\) dollars for each unit. On the other hand, commercial customers pay \(q=1000-y\) dollars for each of the \(y\) units of electricity they use. What price should the power company charge each type of customer to maximize profit? What is the maximum profit?

Cow to Market-A cow currently weighs \(800 \mathrm{lb}\) and is gaining \(35 \mathrm{lb}\) per week. It costs \(\$ 6.50\) a week to maintain the cow. The market price today is \(\$ 0.95\) per pound but is falling \(\$ 0.01\) per day. Formulate a mathematical model and find the optimal period to keep the cow until it is sold to maximize profits.

A constant price has been assumed in all the models developed in this section. Suggest some fisheries for which that assumption is not realistic. How might you alter the assumption? How would you determine the appropriate tax?

Using the basic nonlinear model, \(y=a x^{b}\) fit the following data set and provide the model, a plot of the data and the model, and a residual plot: \begin{tabular}{cc} \hline\(x\) & \(y\) \\ \hline 100 & 150 \\ 125 & 140 \\ 125 & 180 \\ 150 & 210 \\ 150 & 190 \\ 200 & 320 \\ 200 & 280 \\ 250 & 400 \\ 250 & 430 \\ 300 & 440 \\ 300 & 390 \\ 350 & 600 \\ 400 & 610 \\ 400 & 670 \\ \hline \end{tabular}