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College Mathematics for Business Economics Life Sciences and Social Sciences 12th edition Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen - Solutions
A dietitian in a hospital is to arrange a special diet using two foods. Each ounce of food M contains 30 units of calcium, 10 units of iron, and 10 units of vitamin A. Each ounce of food N contains 10 units of calcium, 10 units of iron, and 30 units of vitamin A. The minimum requirements in the
In Problem solve each system of linear inequalities graphically. 3x + 4y ≤ 12 y ≥ -3
In Problem solve each system of linear inequalities graphically. 2x + 5y ≤ 20 x - 5y ≥ -5
Solve the linear programming problems stated in Problem Minimize and maximize P = 3x + 2y subject to 6x + 3y ≤ 24 3x + 6y ≤ 30 x + y ≥ 0
Solve the linear programming problems stated in Problem Minimize and maximize P = 8x + 7y subject to 4x + 3y ≤ 24 3x + 4y ≤ 8 x + y ≤ 0
Solve the linear programming problems stated in Problem Minimize and maximize P = 20x + 10y subject to 3x + y ≤ 21 x + y ≤ 9 x + 3y ≤ 21 x, y ≥ 0
Solve the linear programming problems stated in Problem Minimize and maximize z = 400x + 100y subject to 3x - y ≥ 24 x + 2y ≥ 16 y ≥ 30 x, y ≥ 0
Solve the linear programming problems stated in Problem Minimize and maximize P = 2x + y subject to x + y ≥ 2 6x + 4y ≤ 36 4x + 2y ≤ 20 x, y ≥ 0
In Problem, graph the constant-profit lines through (3, 3) and (6, 6). Use a straightedge to identify the corner point where the maximum profit occurs. Confirm your answer by constructing a corner-point table.P = 4x + y
Solve the linear programming problems stated in Problem Minimize and maximize P = -x + 3y subject to 2x - y ≥ 4 -x + 2y ≤ 4 y ≤ 6 x, y ≥ 0
Solve the linear programming problems stated in Problem Minimize and maximize P = 12x- + 14y subject to -2x: + y ≥ 6 x + y ≤ 15 3x - y ≥ 0 x, y ≥ 0
Solve the linear programming problems stated in Problem Minimize C = 30x + 10y Subject to 1.8x + 0.9y ≥ 270 0.3x + 0.2y ≥ 54 0.01x + 0.03y ≥ 3.9 x, y ≥ 0
Solve the linear programming problems stated in Problem Maximize P = 300x: + 460y subject to 245x- + 452y ≤ 4,181 290x- + 379y ≤ 3,888 390x + 299y ≤ 4,407 x, y ≥ 0
In Problem explain why Theorem 2 cannot be used to conclude that a maximum or minimum value exists. Graph the feasible regions and use graphs of the objective function z = x - y for various values of z to discuss the existence of a maximum value and a minimum value. Minimize and maximize z = x -
The corner points for the feasible region determined by the system of linear inequalities x + 4y ≥ 30 3x- + y ≥ 24 x,y ≥ 0 are A = (0,24), B = (6,6), and D = (30,0). If C = ax + by and a, b >0, determine conditions on a and b that will ensure that the minimum value of C occurs (A) only at
A furniture manufacturing company manufactures dining-room tables and chairs. The relevant manufacturing data are given in the table below.(A) How many tables and chairs should be manufactured each day to realize a maximum profit? What is the maximum profit?(B) Discuss the effect on the production
An electronics firm manufactures two types of personal computers-a standard model and a portable model. The production of a standard computer requires a capital expenditure of $400 and 40 hours of labor. The production of a portable computer requires a capital expenditure of $250 and 30 hours of
If each van can transport 7 people and there are 35 available chaperones, show that the optimal solution found graphically involves decimals. Find all feasible solutions with integer coordinates and identify the one that minimizes the transportation costs. Can this optimal integer solution be
An investor has $24,000 to invest in bonds of AAA and B qualities. The AAA bonds yield an average of 6%, and the B bonds yield 10%. The investor requires that at least three times as much money should be invested in AAA bonds as in B bonds. How much should be invested in each type of bond to
In Problem, graph the constant-profit lines through (3, 3) and (6, 6). Use a straightedge to identify the corner point where the maximum profit occurs. Confirm your answer by constructing a corner-point table.P = 9x + 3y
A fast-food chain plans to expand by opening several new restaurants. The chain operates two types of restaurants, drive-through and full-service. A drive-through restaurant costs $100,000 to construct, requires 5 employees, and has an expected annual revenue of $200,000. A full-service restaurant
A dietitian is to arrange a special diet composed of two foods, M and N. Each ounce of food M contains 30 units of calcium, 10 units of iron, 10 units of vitamin A, and 8 units of cholesterol. Each ounce of food N contains 10 units of calcium, 10 units of iron, 30 units of vitamin A, and 4 units of
A laboratory technician in a medical research center is asked to formulate a diet from two commercially packaged foods, food A and food B, for a group of animals. Each ounce of food A contains 8 units of fat, 16 units of carbohydrate, and 2 units of protein. Each ounce of food B contains 4 units of
A city council voted to conduct a study on inner-city community problems using sociologists and research assistants from a nearby university. Allocation of time and costs per week are given in the table. How many sociologists and how many research assistants should be hired to minimize the cost and
In Problem, graph the constant-cost lines through (9, 9) and (12,12). Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table.C = 7x + 9y
In Problem, graph the constant-cost lines through (9, 9) and (12,12). Use a straightedge to identify the corner point where the minimum cost occurs. Confirm your answer by constructing a corner-point table.C = 2x + 11y
Graph the systems of inequalities in Problems. Introduce slack variables to convert each system of inequalities to a system of equations, and find the basic solution of the system. Construct a table (similar to Table 1) listing basic solution, the corresponding point on the graph, and whether the
Graph the systems of inequalities in Problems. Introduce slack variables to convert each system of inequalities to a system of equations, and find the basic solution of the system. Construct a table (similar to Table 1) listing basic solution, the corresponding point on the graph, and whether the
Repeat Problem 1 for a standard maximization problem with three problem constraints and four decision variables. Problem 1 (A) The number of slack variables that must be introduced to form the system of problem constraint equations (B) The number of basic variables and the number of non-basic
Repeat Problem 3 if the system of problem constraint equations has 10 variables, including 4 slack variables. Problem 3 (A) The number of constraint equations in the system (B) The number of decision variables in the system (C) The number of basic variables and the number of non-basic variables
Repeat Problem 5 for the system 2x1 + x2 + s1 = 30 x1 + 5x2 + s2 = 60
Repeat Problem 7 for the system x1 + 2x2 + s1 = 12 3x1 + 2x2 + s2 = 24
Maximize P = 3x1 + 2x2 Subject to x1 + x2 < 20 x1 + 2x2 < 10 x1, x2 > 0 (A) Using slack variables, write the initial system for each linear programming problem. (B) Write the simplex tableau, circle the first pivot, and identify the entering and exiting variables. (C) Use the simplex method to
Repeat Problem 10 with the objective function changed to P = x1 + 3x2.Problem 10Maximize P = 3x1 + 2x2Subject to x1 + x2 < 20x1 + 2x2 < 10x1, x2 > 0(A) Using slack variables, write the initial system for each linear programming problem.(B) Write the simplex tableau, circle the first pivot,
Solve the linear programming problems in Problems using the simplex method. Maximize P = 15x1 + 20x2 Subject to 2x1 + x2 < 9 x1 + x2 < 6 x1 + 2x2 < 10 x1, x2 > 0
Solve the linear programming problems in Problems using the simplex method. Repeat Problem 15 with P = -x1 + 3x2. In Problem 15 Maximize P = 2x1 + 3x2 Subject to -2x1 + x2 < 2 -x1 + x2 < 5 x2 < 6 x1, x2 > 0
Solve the linear programming problems in Problems using the simplex method. Repeat Problem 17 with P = x1, + 2x2. Problem 17 Maximize P= -x1 + 2x2 Subject to -x1 + x2 < 2 -x1 + 3x2 < 12 x1 - 4x2 < 4 x1, x2 > 0
(A) Identify the basic and non basic variables.
Solve the linear programming problems in Problems using the simplex method. Maximize P = 4x1 - 3x2 + 2x3 Subject to x1 + 2x2 - x3 < 5 3x1 + 2x2 + 2x3 < 22 x1, x2, x3 > 0
Solve the linear programming problem in Problem using the simplex method Maximize subject to P = x1 +x2 +2x3 x1 - 2x2 + x3 ≤ 9 2x1 + x2 + 2x3 ≤ 28 x1,x2,x3 ≥ 0
Solve the linear Programming problem in problem using the simplex method.Repeat problem 25 with P = 20x1 + 20x2.Data from Problem 25Maximize subject toP = 20x1 + 30x20.6x1 + 1.2x2 ≤ 9600.03x1 - 0.04x2 ≤360.3x1 + 0.2x2 ≤ 270x1, x2 ≥ 0
Solve the linear Programming problem in problem using the simplex method. Repeat problem 25 with P = 20x1 + 20x2. Problem 25 Maximize subject to P = 20x1 + 30x2 0.6x1 + 1.2x2 ≤ 960 0.03x1 - 0.04x2 ≤36 0.3x1 + 0.2x2 ≤ 270 x1, x2 ≥ 0
Solve the linear Programming problem in problem using the simplex method. Maximize subject to P = 10x1 + 50x2 + 10x3 3x1 + 3x2 + 3x3 ≤ 66 6x1 - 2x2 + 4x3 ≤48 3x1 + 6x2 + 9x3 ≤ 108 x1, x2, x3 ≥ 0
In Problem, first solve the linear programming problem by the simplex method, keeping track of the basic feasible solutions at each step. Then graph the feasible region and illustrate the path to the optimal solution determined by the simplex method. Maximize subject to P = 5x1 + 3x2 5x1 + 4x2 =s
Solve problem by the simplex method and also by the geometric method. Compare and contrast the result. Maximize Subject to P = 2x1 + 3x2 -x1 + x2 ≤ 2 x2 ≤ 4 x1, x2 ≥
In Problem, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two
In Problem, there is a tie for the choice of the first pivot column. Use the simplex method to solve each problem two different ways: first by choosing column 1 as the first pivot column, and then by choosing column 2 as the first pivot column. Discuss the relationship between these two
Solve Problem 37 with the additional restriction that the combined total number of components produced each week cannot exceed 420. Discuss the effect of this restriction on the solution to Problem 37.
(A) Identify the basic and non basic variables.
Repeat Problem 39 under the additional assumption that no more than $30,000 can be invested in money market funds. Problem 39 An investor has at most $100,000 to invest in government bonds, mutual funds, and money market funds. The average yields for government bonds, mutual funds, and money market
Repeat Problem 41 if the department store increases its budget to $24,000 and requires that at least half of the ads be placed during prime-time.
A company manufactures three-speed, five-speed, and ten-speed bicycles. Each bicycle passes through three departments: fabrication, painting & plating, and final assembly. The relevant manufacturing data are given in the table.How many bicycles of each type should the company manufacture per day in
Repeat Problem 44 if the profit on a five-speed bicycle increases from $70 to $110 and all other data remain the same. If the slack associated with any problem constraint is nonzero, find it.
Repeat Problem 44 if the profit on a five-speed bicycle increases from $70 to $110 and all other data remain the same. If the slack associated with any problem constraint is nonzero, find it.Problem 44A company manufactures three-speed, five-speed, and ten-speed bicycles. Each bicycle passes
Repeat Problem 49 if the scientist wants to maximize the daily calcium intake while not allowing the intake of iron or protein to exceed the average daily intake.Problem 49The natural diet of a certain animal consists of three foods: A, B, and C. The number of units of calcium, iron, and protein in
Repeat Problem 51 if one of the requirements of the grant is that at least 50% of the interviewers be undergraduate students. Problem 51 A political scientist received a grant to fund a research project on voting trends. The budget includes $3,200 for conducting door-to-door interviews on the day
In Problems find the pivot element, identify the entering and exiling variables, and perform one pivot operation.
In Problems find the pivot element, identify the entering and exiling variables, and perform one pivot operation.
In Problems, (A) Form the dual problem. (B) Write the initial system for the dual problem. (C) Write the initial simplex tableau for the dual problem and label the columns of the tableau. Minimize subject to C = 12x1 + 5x2 2x1 + x2 ≥ 7 3x1 + x2 ≥ 9 x1,x2 ≥ 0
In Problem, a minimization problem, the corresponding dual problem, and the final simplex tableau in the solution of the dual problem are given. (A) Find the optimal solution of the dual problem. (B) Find the optimal solution of the minimization problem. Minimize subject to C = 16x1 + 25x2 3x1 +
In Problems , (A) Form the dual problem.
In Problems , (A) Form the dual problem. Discuss.
In Problems , (A) Form the dual problem. Discuss in detail.
In Problems , (A) Form the dual problem. Discuss briefly.
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 2x1 + x2 x1 + x2 ≥ 8 x1 + 2x2 ≥ 4 x1,x2 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 10x1 + 4x2 2x1 + x2 ≥ 6 x1 - 4x2 ≥ -24 -8x1 + 5x2 ≥ -24 x1,x2 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 40x1 + 10x2 3x1 + x2 ≥ 24 x1 + x2 ≥ 16 x1 +4x2 ≥ 30 x1,x2 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 4x1 + 8x2 2x1 + x2 ≥ 12 x1 + x2 ≥ 9 x2 ≥ 4 x1,x2 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 14x1 + 8x2 + 20x3 x1 +x2 + 3x3 ≥ 6 2x1 + x2 - x3 ≥ 9 x1, x2, x3 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem. Minimize subject to C = 6x1 + 8x2 + 3x3 -3x1 - 2x2 + x3 ≥ 4 x1 + x2 - x3 ≥ 2 x1, x2, x3 ≥ 0
Solve the linear programming problem by applying the simplex method to the dual problem.Minimize subject to C = 6x1 + 8x2 + 12x3x1 + 3x2 +3x3 ≥ 6x1 + 5x2 + 5x3 ≥ 42x1 + 2x2 + 3x3 ≥ 8x1 , x2 , x3 ≥ 0
Solve the linear programming problem in problem by applying the simplex method to the dual problem. Repeat Problem 43 with C = 4x1 + 7x2 + 5x3 + 6x4. Problem 43 Minimize C = 5x1 + 4x2 + 5x3 + 6x4 subject to x1 + x2 ≤ 12 x3 + x4 ≤ 25 x1 + x3 ≥ 20 x2 + x4 ≥ 15 x1, x2, x3, x4 ≥ 0
A mining company operates two mines, each producing three grades of ore. The West Summit mine can produce 2 tons of low-grade ore, 3 tons of medium-grade ore, and 1 ton of high-grade ore in one hour of operation. The North Ridge mine can produce 2 tons of low-grade ore, 1 ton of medium-grade ore,
Repeat Problem 46 if it costs $300 per hour to operate the West Summit mine and $700 per hour to operate the North Ridge mine and all other data remain the same. Problem 46 A mining company operates two mines, each producing three grades of ore. The West Summit mine can produce 2 tons of low-grade
Repeat Problem 46 if it costs $800 per hour to operate the West Summit mine and $200 per hour to operate the North Ridge mine and all other data remain the same. Problem 46 A mining company operates two mines, each producing three grades of ore. The West Summit mine can produce 2 tons of low-grade
A farmer can buy three types of plant food: mix A, mix B, and mix C. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 10 pounds of nitrogen, and 10 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 10 pounds of nitrogen, and 15 pounds of potash. Each
Repeat Problem 53 if the weekly cost of busing a student from North Division to Washington is $7 and all other data remain the same.Problem 53A metropolitan school district has two overcrowded high schools and two under enrolled high schools. To balance the enrollment, the school board decided to
In Problem, find the transpose of each matrix.
In Problem, find the transpose of each matrix.
Use the big M method to solve Problem. P= -4x1+ 16x2 3x1 + x2 ≤ 28 x1 + 2x2 ≥ 16 x1, x2 ≥ 0
Use the big M method to solve Problem. P= -6x1+ 2x2 x1 + 2x2 ≤ 20 2x1 + x2 ≤ 16 x1 + x2 ≥ 9 x1, x2 ≥0
Use the big M method to solve Problem. P= 5x1+ 7x2 - 9x3 x1 - x2 + x3 ≥ 20 x1 + x2 + 5x3 = 35 x1, x2 , x3 ≥ 0
Use the big M method to solve Problem. C= -3x1+ 15x2 - 4x3 2x1 + x2 + 3x3 ≤ 24 x1 + 2x2 + x3 ≥ 6 x1 - 3x2 + x3 = 2 x1, x2 , x3 ≥ 0
Use the big M method to solve Problem. P=3x1+ 6x2 + 2x3 2x1 + 2x2 + 3x3 ≤ 12 2x1 + 2x2 + x3 = 90 x1, x2 , x3 ≥ 0
In Problem,(1) Introduce slack, surplus, and artificial variables and form the modified problem.(2) Write the preliminary simplex tableau for the modified problem and find the initial simplex tableau.(3) Find the optimal solution of the modified problem by applying the simplex method to the initial
Use the big M method to solve Problem. P=5x1+ 2x2 + 9x3 2x1 + 4x2 + x3 ≤ 150 3x1 + 3x2 + x3 ≤ 90 -x1 + 5x2 + x3 ≥ 120 x1, x2 , x3 ≥ 0
Using the big M method to solve Problem Maximize subject to P=2x1+ 4x2 + x3 2x1 + 3x1 + 5x3 ≤ 280 2x1 + 2x2 + x3 ≥140 2x1 + x2 ≥ 150 x1,x2,x3 ≥ 0
Repeat Problem 23 with Problem 6 and 8 Problem 23
Minimize subject to P = 7x1 - 5x2 + 2x3 x1 - 2x2 - x3 ≥ -8 x1 - x2 + x3 ≤ 10 x1, x2, x3, ≥ 0 Problem are mixed. Some can be solved by the methods presented in Sections 6-2 and 6-3, while others must be solved by the big M method.
Minimize subject to C = -5x1 + 10x2 - 15x3 2x1 + 3x2 - x3 ≤ 24 x1 - 2x2 - 2x3 ≥ 1 x1, x2 , x3 ≥ 0 Problem are mixed. Some can be solved by the methods presented in Sections 6-2 and 6-3, while others must be solved by the big M method.
Minimize subject to P = 8x1 + 2x2 - 10x3 x1 + x2 - 3x3 ≤ 20 4x1 - x2 - 2x3 ≤ - 7 x1, x2 , x3 ≥ 0 Problem are mixed. Some can be solved by the methods presented in Sections 6-2 and 6-3, while others must be solved by the big M method.
Discuss the effect on the solution to Problem 33 if the Tribune will not accept more than 4 ads from the company. In Problems 33-38, construct a mathematical model in the form of a linear programming problem. Then solve the problem using the big M method.
Discuss the effect on the solution to Problem 35 if the cost of brand C liquid diet food increases to $1.50 per bottle. In Problem, construct a mathematical model in the form of a linear programming problem. Then solve the problem using the big M method.
Discuss the effect on the solution to Problem 37 if the limit on phosphoric acid is increased to 1,000 pounds. In Problem, construct a mathematical model in the form of a linear programming problem. Then solve the problem using the big M method.
In Problem, (A) Introduce slack, surplus, and artificial variables and form the modified problem. (B) Write the preliminary simplex tableau for the modified problem and find the initial simplex tableau.
A savings and loan company has $3 million to lend. The types of loans and annual returns offered are given in the table. State laws require that at least 50% of the money loaned for mortgages must be for first mortgages and that at least 30% of the total amount loaned must be for either first or
A company makes two brands of trail mix, regular and deluxe, by mixing dried fruits, nuts, and cereal. The recipes for the mixes are given in the table. The company has 1,200 pounds of dried fruits, 750 pounds of nuts, and 1,500 pounds of cereal for the mixes. The company makes a profit of $0.40 on
Refer to Problem 43. Suppose the investor decides that she would like to minimize the total risk factor, as long as her return does not fall below 9%. What percentage of her total investments should be invested in each choice to minimize the total risk level? In Problem, construct a mathematical
A farmer grows three crops: corn, oats, and soybeans. He mixes them to feed his cows and pigs. At least 40% of the feed mix for the cows must be corn. The feed mix for the pigs must contain at least twice as much soybeans as corn. He has harvested 1,000 bushels of corn, 500 bushels of oats, and
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