Exam 3: Linear Programming: a Geometric Approach

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 lb of cast iron and 6 min of labor. To produce each model B grate requires 6 lb of cast iron and 3 min of labor. The profit for each model A grate is $2, and the profit for each model B grate is $1.50. 1,170 lb of cast iron and 21 labor-hours are available for the production of grates each day. Because of an excess inventory of model A grates, management has decided to limit the production of model A grates to no more than 210 grates per day. How many grates of each model should the division produce daily to maximize Kane's profits? ​ Find the range of values that the resource for cast iron can assume without changing the optimal solution. Find the shadow price for the resource for cast iron. Round your answer to the nearest cent. ​

AntiFam, a hunger-relief organization, has earmarked between $4 and $4.5 million, inclusive, for aid to two African countries, country A and country B. Country A is to receive between $3 and $3.5 million, inclusive, in aid, and country B is to receive at least $0.75 million in aid. It has been estimated that each dollar spent in country A will yield an effective return of $0.60, whereas a dollar spent in country B will yield an effective return of $0.70. How should the aid be allocated if the money is to be utilized most effectively according to these criteria? Hint: If and denote the amount of money (in million of dollars) to be given to country A and country B, respectively, then the objective function to be maximized is .

Determine graphically the solution set for the system of inequalities and indicate whether the solution set is bounded or unbounded. ​ ​

Find the graphical solution of the inequality. ​ ​

A corporation has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $2.50, and the profit for each model B grate is $1.50. If 1,000 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 help maximize the division's profits?

Write a system of linear inequalities that describes the shaded region.

Write a system of linear inequalities that describes the shaded region. ​

Find the graphical solution of the inequality. ​ ​

Ace Novelty manufactures Gaint Pandas and Saint Bemards. Each Panda requires yd² of plush, ft³ of stuffing, and pieces of trim; each Saint Bernard requires yd²of plush, ft³ of stuffing, pieces of trim. The profit for each Panda is and profit for each Saint Bernard . If yd² of plush, ft³ of stuffing and pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize the profit?

Write a system of linear inequalities that describes the shaded region. Answer or .

You are given a linear programming problem. Find the shadow price for requirement 1. Maximize

Write a system of linear inequalities that describes the shaded region. ​ ​ Answer or . ​

A corporation has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $3.00, and the profit for each model B grate is $2.50. If 1,000 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 help maximize the division's profits? __________ model A, __________ model B What is the optimal profit? $__________

Ashley has earmarked at most $270,000 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% per year, the international equity fund has a rate of return of 10% per year, and the growth-and-income fund has a rate of return of 16% per 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?

Find the graphical solution of the inequality. ​ ​

Write a system of linear inequalities that describes the shaded region. ​

Determine graphically the solution set for the system of inequalities and indicate whether the solution set is bounded or unbounded. ​ ​

Find the optimal (maximum and/or minimum) value(s) of the given objective function on the feasible set S. If no maximum or minimum value exists, enter no. Z = 8x + 6y Max.: __________, min.: __________

Determine graphically the solution set for the system of inequalities and indicate whether the solution set is bounded or unbounded.

You are given a linear programming problem.Use the method of corners to solve the problem. Maximize