Explanation Given: The laboratory purchases feed A and B for its animals. Feed A contains 2 units of carbohydrate and 4 units of protein per pound. Feed B has 8 units and 2 units of carbohydrate and protein respectively. The cost of Feed A and B are $1.40 and $1.60 respectively. Calculation: Let x denote the number of units of Feed A and y denotes the number of units of Feed B. Given that, at least 80 units of carbohydrate and 132 units of protein are required. The amount of carbohydrate in feed A is 2 units and feed B is 8 units per pound. The required amount is at least 80 units. Therefore, the statement can be written as an inequality, 2 x + 8 y ≥ 80 . Also, the amount of protein in feed A is 4 units and feed B is 2 units per pound. The required amount is at least 132 units which can be written as the inequality 4 x + 2 y ≥ 132 . Since the number of pounds is positive, the value x ≥ 0 and y ≥ 0 . Therefore, the inequality that describes the requirement is { 2 x + 8 y ≥ 80 4 x + 2 y ≥ 132 x ≥ 0 , y ≥ 0 . Solve the equations in terms of y . 2 x + 8 y ≥ 80 8 y ≥ 80 − 2 x y ≥ 80 − 2 x 8 Also, 4 x + 2 y ≥ 132 2 y ≥ 132 − 4 x y ≥ 132 − 4 x 2 y ≥ 66 − 2 x Thus, the inequalities can be rewritten as, { y ≥ 80 − 2 x 8 y ≥ 66 − 2 x x ≥ 0 , y ≥ 0 (1) The cost of feed A per pound is $1.40 and the cost of feed B is $1.60. Thus, the objective function to be minimized is f = 1.40 x + 1.60 y . Sketch the graph of the inequalities (1) and identify the corners of the region. The boundary equations of the above inequalities are , { y = 80 − 2 x 8 y = 66 − 2 x x = 0 , y = 0 (2) Sketch the graph of the straight lines y = 80 − 2 x 8 , y = 66 − 2 x , x = 0 and y = 0 as shown below in Figure 1. From Figure 1, it is observed that the system (2) divides the plane into 4 parts. The solution region of the given inequality set is obtained as follows. Take any arbitrary test point in the region I, to test the solution region of the system (1). Let ( 0 , 40 ) be the test point of the region I. Substitute x = 0 and y = 40 in the system (1) to obtain the system, { 40 ≥ 80 − 2 ( 0 ) 8 40 ≥ 66 − 2 ( 0 ) 0 ≥ 0 , 40 ≥ 0 That is, the simplified inequality is { 40 ≥ 10 40 ≥ 66 0 ≥ 0 , 40 ≥ 0 Since the expression 40 ≥ 66 is not true, the point ( 0 , 40 ) does not satisfy the system (1). Thus, the region I is not the solution region of the system (1). Take any arbitrary test point in the region II, to test the solution region of the system (1). Let ( 40 , 40 ) be the test point of the region I. Substitute x = 40 and y = 40 in the system (1) to obtain the system, { 40 ≥ 80 − 2 ( 40 ) 8 40 ≥ 66 − 2 ( 40 ) 40 ≥ 0 , 40 ≥ 0 That is, the simplified inequality is { 40 ≥ 0 40 ≥ − 14 40 ≥ 0 , 40 ≥ 0 Therefore the point is ( 40 , 40 )

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Chapter 8, Problem 46RE

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Chapter 8, Problem 45RE

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Solve the linear system of equations attached using Gaussian elimination (not Gauss-Jordan) and back subsitution. Remember that: A matrix is in row echelon form if Any row that consists only of zeros is at the bottom of the matrix. The first non-zero entry in each other row is 1. This entry is called aleading 1. The leading 1 of each row, after the first row, lies to the right of the leading 1 of the previous row.

Chapter 8 Solutions

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Ch. 8.1 - In Exercises 16, graph each inequality. 1. y 5x ...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - In Exercises 710, match each solution of a system...Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - In Exercises 1116, the graph of the boundary...Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - For each system of inequalities in Exercises 1722,...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 25E Ch. 8.1 - In Exercises 2326, graph the solution of the...Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Sales Trix Auto Sales sells used cars. To promote...Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Manufacturing A company manufactures two types of...Ch. 8.1 - Prob. 34E Ch. 8.1 - Production A firm produces three different-size...Ch. 8.1 - Advertising Tire Town is developing an advertising...Ch. 8.1 - Nutrition A privately owned lake contains two...Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Manufacturing Safeco Company produces two types of...Ch. 8.1 - Manufacturing A company manufactures commercial...Ch. 8.1 - Constructing a Box An open box with a square base...Ch. 8.2 - In Exercises 1-4, use the given feasible region...Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Find the maximum possible value of f = 400x + 300y...Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Find the maximum possible value of f = 80x + 160y...Ch. 8.2 - Prob. 12E Ch. 8.2 - Find the minimum possible value of g = 40x + 30y...Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Manufacturing A company manufactures two types of...Ch. 8.2 - Manufacturing Evergreen Company produces two types...Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Politics A political candidate wishes to use a...Ch. 8.2 - Prob. 31E Ch. 8.2 - Manufacturing Kitchen Pride manufactures toasters...Ch. 8.3 - Prob. 1E Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Find the first 5 terms of the sequence defined by...Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Salaries Suppose you are offered a job with a...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Bouncing Ball A ball is dropped from 128 feet. 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The number of pounds of feed required to minimize the cost and find the minimum cost.

Solution Summary: The author explains that the laboratory purchases feed A and B for its animals. The required amount of carbohydrate and protein is at least 80 units, which can be written as an inequality.

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Chapter 8 Solutions