Chapter 6

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x+y & \geq 20 \\ x+2 y & \geq 40 \\ x \geq 0, y & \geq 0 \end{aligned} $$

National Business Machines Corporation manufactures two models of fax machines: \(A\) and \(B\). Each model A costs $$\$ 100$$ to make, and each model B costs $$\$ 150$$. The profits are $$\$ 30$$ for each model-A and $$\$ 40$$ for each model-B fax machine. If the total number of fax machines demanded each month does not exceed 2500 and the company has earmarked no more than $$\$ 600,000 /$$ month for manufacturing costs, find how many units of each model National should make each month in order to maximize its monthly profit. What is the largest monthly profit the company can make?

A company manufactures two products, \(\mathrm{A}\) and \(\mathrm{B}\), on two machines, 1 and II. It has been determined that the company will realize a profit of $$\$ 3 $$ unit of product \(A\) and a profit of $$\$ 4 $$ unit of product \(\mathrm{B}\). To manufacture a unit of product A requires 6 min on machine \(\mathrm{I}\) and 5 min on machine II. To manufacture a unit of product \(\mathrm{B}\) requires \(9 \mathrm{~min}\) on machine \(\mathrm{I}\) and 4 min on machine II. There are \(5 \mathrm{hr}\) of machine time available on machine I and \(3 \mathrm{hr}\) of machine time available on machine II in each work shift. How many units of each product should be produced in each shift to maximize the company's profit? What is the optimal profit?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem Maximize \(\quad P=x y\) $$ \text { subject to } \begin{aligned} & 2 x+3 y \leq 12 \\ & 2 x+y \leq 8 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} 3 x-7 y & \geq-24 \\ x+3 y & \geq 8 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model-A hibachi requires \(3 \mathrm{lb}\) of cast iron and \(6 \mathrm{~min}\) of labor. To produce each model-B hibachi requires \(4 \mathrm{lb}\) of cast iron and \(3 \mathrm{~min}\) of labor. The profit for each modelA hibachi is $$\$ 2$$, and the profit for each model-B hibachi is $$\$ 1.50 .$$ If \(1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of hibachis each day, how many hibachis of each model should the division produce in order to maximize Kane's profit? What is the largest profit the company can realize? Is there any raw material left over?

National Business Machines manufactures two models of fax machines: A and B. Each model A costs $$\$ 100$$ to make, and each model \(\mathrm{B}\) costs $$\$ 150$$. The profits are $$\$ 30$$ for each model \(\mathrm{A}\) and $$\$ 40$$ for each model B fax machine. If the total number of fax machines demanded per month does not exceed 2500 and the company has earmarked no more than $$\$ 600,000 $$ month for manufacturing costs, how many units of each model should National make each month in order to maximize its monthly profit? What is the optimal profit?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The problem $$ \begin{aligned} \text { Minimize } & C=2 x+3 y \\ \text { subject to } & 2 x+3 y \leq 6 \\ & x-y=0 \\ & x \geq 0, y \geq 0 \end{aligned} $$ is a linear programming problem.

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} 3 x+4 y & \geq 12 \\ 2 x-y & \geq-2 \\ 0 \leq y & \leq 3 \\ x & \geq 0 \end{aligned} $$

A farmer has 150 acres of land suitable for cultivating crops \(A\) and \(B\). The cost of cultivating crop \(A\) is $$\$40$$/acre whereas that of crop \(B\) is $$\$60$$/acre. The farmer has a maximum of $$\$ 7400$$ available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 laborhours. The farmer has a maximum of 3300 labor-hours available. If he expects to make a profit of $$\$ 150$$ /acre on crop \(\mathrm{A}\) and $$\$ 200$$ /acre on crop \(\mathrm{B}\), how many acres of each crop should he plant in order to maximize his profit? What is the largest profit the farmer can realize? Are there any resources left over?

Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires \(3 \mathrm{lb}\) of cast iron and \(6 \mathrm{~min}\) of labor. To produce each model B grate requires \(4 \mathrm{lb}\) of cast iron and 3 min of labor. The profit for each model A grate is $$\$ 2.00$$, and the profit for each model B grate is $$\$ 1.50$$. If \(1000 \mathrm{lb}\) of cast iron and 20 labor-hours are available for the production of fireplace grates per day, how many grates of each model should the division produce in order to maximize Kane's profit? What is the optimal profit?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{array}{rr} x+2 y \geq & 3 \\ 5 x-4 y \leq & 16 \\ 0 \leq y \leq & 2 \\ x & \geq 0 \end{array} $$

A financier plans to invest up to $$\$ 500,000$$ in two projects. Project A yields a return of \(10 \%\) on the investment whereas project \(\bar{B}\) yields a return of \(15 \%\) on the investment. Because the investment in project \(\mathrm{B}\) is riskier than the investment in project \(\mathrm{A}\), the financier has decided that the investment in project \(\mathrm{B}\) should not exceed \(40 \%\) of the total investment. How much should she invest in each project in order to maximize the return on her investment? What is the maximum return?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x+y & \leq 4 \\ 2 x+y & \leq 6 \\ 2 x-y & \geq-1 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Ashley has earmarked at most $$\$ 250,000$$ for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and- income fund. The money market fund has a rate of return of \(6 \% / y\) ear, the international equity fund has a rate of return of \(10 \% /\) year, and the growth-andincome fund has a rate of return of \(15 \% /\) year. Ashley has stipulated that no more than \(25 \%\) of her total portfolio should be in the growth-and-income fund and that no more than \(50 \%\) of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $$\$ 45$$, and the profit for each chair is $$\$ 20 .$$ In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture in order to maximize its profit? What is the maximum profit?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} 6 x+5 y & \leq 30 \\ 3 x+y & \geq 6 \\ x+y & \geq 4 \\ x \geq 0, y & \geq 0 \end{aligned} $$

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $$\$ 45$$, and the profit for each chair is $$\$ 20$$. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture in order to maximize its profit? What is the maximum profit?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} 6 x+7 y & \leq 84 \\ 12 x-11 y & \leq 18 \\ 6 x-7 y & \leq 28 \\ x \geq 0, y & \geq 0 \end{aligned} $$

A company manufactures products \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Each product is processed in three departments: I, II, and III. The total available labor-hours per week for departments I, II, and III are 900,1080 , and 840 , respectively. The time requirements (in hours per unit) and profit per unit for each product are as follows: $$ \begin{array}{lccc} \hline & \text { Product A } & \text { Product B } & \text { Product C } \\ \hline \text { Dept. I } & 2 & 1 & 2 \\ \hline \text { Dept. II } & 3 & 1 & 2 \\ \hline \text { Dept. III } & 2 & 2 & 1 \\ \hline \text { Profit } & \$ 18 & \$ 12 & \$ 15 \\ \hline \end{array} $$ How many units of each product should the company produce in order to maximize its profit? What is the largest profit the company can realize? Are there any resources left over?

Madison Finance has a total of $$\$ 20$$ million earmarked for homeowner and auto loans. On the average, homeowner loans have a \(10 \%\) annual rate of return, whereas auto loans yield a \(12 \%\) annual rate of return. Management has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans. Determine the total amount of loans of each type that Madison should extend to each category in order to maximize its returns. What are the optimal returns?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x-y & \geq-6 \\ x-2 y & \leq-2 \\ x+2 y & \geq 6 \\ x-2 y & \geq-14 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Ace Novelty manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires \(1.5 \mathrm{yd}^{2}\) of plush, \(30 \mathrm{ft}^{3}\) of stuffing, and 5 pieces of trim; each Saint Bernard requires \(2 \mathrm{yd}^{2}\) of plush, \(35 \mathrm{ft}^{3}\) of stuffing, and 8 pieces of trim. The profit for each Panda is $$\$ 10$$, and the profit for each Saint Bernard is $$\$ 15$$. If \(3600 \mathrm{yd}^{2}\) of plush, \(66,000 \mathrm{ft}^{3}\) of stuffing, and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize its profit? What is the maximum profit?

A financier plans to invest up to $$\$ 500,000$$ in two projects. Project A yields a return of \(10 \%\) on the investment whereas project \(\mathrm{B}\) yields a return of \(15 \%\) on the investment. Because the investment in project \(\mathrm{B}\) is riskier than the investment in project \(\mathrm{A}\), the financier has decided that the investment in project \(B\) should not exceed \(40 \%\) of the total investment. How much should she invest in each project in order to maximize the return on her investment? What is the maximum return?

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded. $$ \begin{aligned} x-3 y & \geq-18 \\ 3 x-2 y & \geq 2 \\ x-3 y & \leq-4 \\ 3 x-2 y & \leq 16 \\ x \geq 0, y & \geq 0 \end{aligned} $$

As part of a campaign to promote its annual clearance sale, Excelsior Company decided to buy television advertising time on Station KAOS. Excelsior's television advertising budget is $$\$ 102,000$$. Morning time costs $$\$ 3000$$ /minute, afternoon time costs $$\$ 1000$$/ minute, and evening (prime) time costs $$\$ 12,000 /$$ minute. Because of previous commitments, KAOS cannot offer Excelsior more than \(6 \mathrm{~min}\) of prime time or more than a total of \(25 \mathrm{~min}\) of advertising time over the 2 weeks in which the commercials are to be run. KAOS estimates that morning commercials are seen by 200,000 people, afternoon commercials are seen by 100,000 people, and evening commercials are seen by 600,000 people. How much morning, afternoon, and evening advertising time should Excelsior buy to maximize exposure of its commercials?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Sharon has a total of $$\$ 200,000$$ to invest in three types of mutual funds: growth, balanced, and income funds. Growth funds have a rate of return of \(12 \% /\) year, balanced funds have a rate of return of \(10 \% /\) year, and income funds have a return of \(6 \%\) /year. The growth, balanced, and income mutual funds are assigned risk factors of \(0.1,0.06\), and \(0.02\), respectively. Sharon has decided that at least \(50 \%\) of her total portfolio is to be in income funds and at least \(25 \%\) in balanced funds. She has also decided that the average risk factor for her investment should not exceed \(0.05\). How much should Sharon invest in each type of fund in order to realize a maximum return on her investment? What is the maximum return?

A farmer plans to plant two crops, A and \(\mathrm{B}\). The cost of cultivating crop \(\mathrm{A}\) is $$\$ 40$$ acre whereas that of crop B is $$\$60/acre$$. The farmer has a maximum of $$\$ 7400$$ available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop \(\mathrm{B}\) requires 25 labor-hours. The farmer has a maximum of 3300 labor-hours available. If she expects to make a profit of $$\$ 150 $$ acre on crop \(A\) and $$\$ 200$$ acre on crop \(B\), how many acres of each crop should she plant in order to maximize her profit? What is the optimal profit?

Custom Office Furniture is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and \(\mathrm{C}\) -are to be marketed. Each model-A desk requires \(1 \frac{1}{4} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each model-B desk requires \(1 \frac{1}{2} \mathrm{hr}\) for fabrication, \(1 \mathrm{hr}\) for assembly, and \(1 \mathrm{hr}\) for finishing; each modelC desk requires \(1 \frac{1}{2} \mathrm{hr}, \frac{3}{4} \mathrm{hr}\), and \(\frac{1}{2} \mathrm{hr}\) for fabrication, assembly, and finishing, respectively. The profit on each model-A desk is $$\$ 26$$, the profit on each model-B desk is $$\$ 28$$, and the profit on each model-C desk is $$\$ 24$$. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is \(310 \mathrm{hr}, 205 \mathrm{hr}\), and \(190 \mathrm{hr}\), respectively. To maximize Custom's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?

Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $$\$ 14,000 $$ day to operate, and it yields 50 oz of gold and 3000 oz of silver each day. The Horseshoe Mine costs $$\$ 16,000 $$ day to operate, and it yields 75 oz of gold and 1000 oz of silver each day. Company management has set a target of at least 650 oz of gold and 18,000 oz of silver. How many days should each mine be operated so that the target can be met at a minimum cost? What is the minimum cost?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle.

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 60 deluxe cabins and 160 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 -day cruise in May. It costs $$\$ 44,000$$ to operate a type-A vessel and $$\$ 54,000$$ to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

The water-supply manager for a Midwest city needs to supply the city with at least 10 million gal of potable (drinkable) water per day. The supply may be drawn from the local reservoir or from a pipeline to an adjacent town. The local reservoir has a maximum daily yield of 5 million gal of potable water, and the pipeline has a maximum daily yield of 10 million gallons. By contract, the pipeline is required to supply a minimum of 6 million gallons/day. If the cost for 1 million gallons of reservoir water is $$\$ 300$$ and that for pipeline water is $$\$ 500$$, how much water should the manager get from each source to minimize daily water costs for the city? What is the minimum daily cost?

Company has decided to introduce three fruit juices made from blending two or more concentrates. These juices will be packaged in 2-qt (64-oz) cartons. One carton of pineapple-orange juice requires 8 oz each of pineapple and orange juice concentrates. One carton of orange-banana juice requires \(12 \mathrm{oz}\) of orange juice concentrate and 4 oz of banana pulp concentrate. Finally, one carton of pineapple-orange-banana juice requires 4 oz of pineapple juice concentrate, 8 oz of orange juice concentrate, and 4 oz of banana pulp. The company has decided to allot 16,000 oz of pineapple juice concentrate, 24,000 oz of orange juice concentrate, and 5000 oz of banana pulp concentrate for the initial production run. The company has also stipulated that the production of pineappleorange-banana juice should not exceed 800 cartons. Its profit on one carton of pineapple-orange juice is $$\$ 1.00$$, its profit on one carton of orange-banana juice is $$\$ .80$$, and its profit on one carton of pineapple-orange-banana juice is $$\$ .90$$. To realize a maximum profit, how many cartons of each blend should the company produce? What is the largest profit it can realize? Are there any concentrates left over?

Ace Novelty manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires \(1.5 \mathrm{yd}^{2}\) of plush, \(30 \mathrm{ft}^{3}\) of stuffing, and 5 pieces of trim; each Saint Bernard requires \(2 \mathrm{yd}^{2}\) of plush, \(35 \mathrm{ft}^{3}\) of stuffing, and 8 pieces of trim. The profit for each Panda is $$\$ 10$$, and the profit for each Saint Bernard is $$\$ 15$$. If \(3600 \mathrm{yd}^{2}\) of plush, \(66,000 \mathrm{ft}^{3}\) of stuffing and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize profit? What is the maximum profit?

A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. She has decided that the meals should contain a minimum of \(400 \mathrm{mg}\) of calcium, \(10 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin C. She has further decided that the meals are to be prepared from foods \(\mathrm{A}\) and \(\mathrm{B}\). Each ounce of food \(\mathrm{A}\) contains \(30 \mathrm{mg}\) of calcium, \(1 \mathrm{mg}\) of iron, \(2 \mathrm{mg}\) of vitamin \(\mathrm{C}\), and \(2 \mathrm{mg}\) of cholesterol. Each ounce of food \(\mathrm{B}\) contains \(25 \mathrm{mg}\) of calcium, \(0.5 \mathrm{mg}\) of iron, \(5 \mathrm{mg}\) of vitamin \(\mathrm{C}\), and \(5 \mathrm{mg}\) of cholesterol. Find how many ounces of each type of food should be used in a meal so that the cholesterol content is minimized and the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met.

Consider the linear programming problem $$ \begin{array}{lr} \text { Maximize } & P=3 x+2 y \\ \text { subject to } & x-y \leq 3 \\ x & \leq 2 \\ x \geq 0, y & \geq 0 \end{array} $$ a. Sketch the feasible set for the linear programming problem. b. Show that the linear programming problem is unbounded. c. Solve the linear programming problem using the \(\operatorname{sim}\) plex method. How does the method break down? d. Explain why the result in part (c) implies that no solution exists for the linear programming problem.

AntiFam, a hunger-relief organization, has earmarked between $$\$ 2$$ and $$\$ 2.5$$ million (inclusive) for aid to two African countries, country \(\mathrm{A}\) and country B. Country \(\mathrm{A}\) is to receive between $$\$ 1$$ million and $$\$ 1.5$$ million (inclusive), and country \(B\) is to receive at least $$\$ 0.75$$ million. It has been estimated that each dollar spent in country A will yield an effective return of $$\$ .60$$, whereas a dollar spent in country \(\mathrm{B}\) will yield an effective return of $$\$ .80 .$$ How should the aid be allocated if the money is to be utilized most effectively according to these criteria? Hint: If \(x\) and \(y\) denote the amount of money to be given to country A and country B, respectively, then the objective function to be maximized is \(P=0.6 x+0.8 y\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.

Travel has decided to advertise in the Sunday editions of two major newspapers in town. These advertisements are directed at three groups of potential customers. Each advertisement in newspaper I is seen by 70,000 group-A customers, 40,000 group-B customers, and 20,000 group-C customers. Each advertisement in newspaper II is seen by 10,000 group-A, 20,000 group-B, and 40,000 group-C customers. Each advertisement in newspaper I costs $$\$ 1000$$, and each advertisement in newspaper II costs $$\$ 800$$. Everest would like their advertisements to be read by at least 2 million people from group A. \(1.4\) million people from group \(\mathrm{B}\), and 1 million people from group C. How many advertisements should Everest place in each newspaper to achieve its advertising goals at a minimum cost? What is the minimum cost?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Choosing the pivot row by requiring that the ratio associated with that row be the smallest ensures that the iteration will not take us from a feasible point to a nonfeasible point.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Choosing the pivot column by requiring that it be the column associated with the most negative entry to the left of the vertical line in the last row of the simplex tableau ensures that the iteration will result in the greatest increase or, at worse, no decrease in the objective function.

Bata Aerobics manufactures two models of steppers used for aerobic exercises. Manufacturing each luxury model requires \(10 \mathrm{lb}\) of plastic and \(10 \mathrm{~min}\) of labor. Manufacturing each standard model requires \(16 \mathrm{lb}\) of plastic and \(8 \mathrm{~min}\) of labor. The profit for each luxury model is $$\$ 40$$, and the profit for each standard model is $$\$ 30 .$$ If \(6000 \mathrm{lb}\) of plastic and 60 laborhours are available for the production of the steppers per day, how many steppers of each model should Bata produce each day in order to maximize its profit? What is the optimal profit?

Patricia has at most $$\$ 30,000$$ to invest in securities in the form of corporate stocks. She has narrowed her choices to two groups of stocks: growth stocks that she assumes will yield a \(15 \%\) return (dividends and capital appreciation) within a year and speculative stocks that she assumes will yield a \(25 \%\) return (mainly in capital appreciation) within a year. Determine how much she should invest in each group of stocks in order to maximize the return on her investments within a year if she has decided to invest at least 3 times as much in growth stocks as in speculative stocks.

A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than \(8 \mathrm{oz}\) and must contain at least 29 units of nutrient \(I\) and 20 units of nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: brand \(A\) and brand \(B\). Each ounce of brand A contains 3 units of nutrient \(I\) and 4 units of nutrient II. Each ounce of brand B contains 5 units of nutrient I and 2 units of nutrient II. Brand A costs 3 cents/ounce and brand B costs 4 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the Ti

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution.

Consider the linear programming problem $$ \begin{array}{lr} \text { Maximize } & P=2 x+7 y \\ \text { subject to } & 2 x+y \geq 8 \\ x+y & \geq 6 \\ x & \geq 0, y \geq 0 \end{array} $$ a. Sketch the feasible set \(S\). b. Find the corner points of \(S\). c. Find the values of \(P\) at the corner points of \(S\) found in part (b). d. Show that the linear programming problem has no (optimal) solution. Does this contradict Theorem \(1 ?\)

Consider the linear programming problem $$ \begin{aligned} \text { Minimize } & C=-2 x+5 y \\ \text { subject to } & x+y \leq 3 \\ & 2 x+y \leq 4 \\ & 5 x+8 y \geq 40 \\ & x \geq 0, y \geq 0 \end{aligned} $$ a. Sketch the feasible set. b. Find the solution(s) of the linear programming problem, if it exists.