Explanation Given: One unit of plants and one animal provides 30 units of energy and 20 units of energy. The plants and animals need at least 360 units of energy. One unit of plants and animals provides 10 units of protein and 25 units of protein. The plants and animals need at least 300 units of protein. One unit of plants and one animal provides no hides and 1 hide respectively. The plants and animals need at least 8 hides. The available units of plants are 25. That is x ≤ 25 . The available animals are 25. One unit of a plant requires 30 hour and one animal requires 15 hour for the cost of the population. Calculation: Let x and y represents the number of units of plants and animals, respectively. From the given information, the linear programing problem can be stated as follows, Minimize z = 30 x + 15 y Subject to the constraints, 30 x + 20 y ≥ 360 (1) 10 x + 25 y ≥ 300 (2) y ≥ 8 (3) x ≤ 25 (4) y ≤ 25 (5) x ≥ 0 y ≥ 0 Consider the equation (1). The given inequality in the equation is ≥ . Therefore, the solid line is used to join the coordinates of this equation. Now, replace the symbol ≥ by the symbol = in given equation (1). Then the equation (1) becomes, 30 x + 20 y = 360 . Obtain any 2 coordinates of the above equation to plot on the line. Consider x = 0 , 12 and obtain the corresponding values of y respectively. Substitute x = 0 in the linear equation 30 x + 20 y = 360 . 30 ( 0 ) + 20 y = 360 20 y = 360 y = 360 20 y = 18 Thus, the value of y is 18 at x = 0 . Hence, the first coordinate is ( 0 , 18 ) . Substitute x = 12 in the linear equation 30 x + 20 y = 360 . 30 ( 12 ) + 20 y = 360 360 + 20 y = 360 20 y = 360 − 360 y = 0 Hence, the second coordinate is ( 12 , 0 ) . Therefore, the coordinates are ( 0 , 18 ) and ( 12 , 0 ) . Use the test point to determine the half plane which is to be shaded. Let the test point be ( 0 , 0 ) . Now substitute the test point ( 0 , 0 ) in 30 x + 20 y ≥ 360 . 30 ( 0 ) + 20 ( 0 ) ≥ 360 0 ≥ 360 It is noticed that the inequality 30 x + 20 y ≥ 360 is false. Thus, shade the half-plane that does not contain the test point ( 0 , 0 ) . That is, the half plane above the line. Consider the equation (2). The given inequality in the equation is ≥ . Therefore, the solid line is used to join the coordinates of this equation. Now, replace the symbol ≥ by the symbol = in given equation (2). Then the equation (2) becomes, 10 x + 25 y = 300 . Obtain any 2 coordinates of the above equation to plot on the line. Consider x = 0 , 30 and obtain the corresponding values of y respectively. Substitute x = 0 in the linear equation 10 x + 25 y = 300 . 10 ( 0 ) + 25 y = 300 25 y = 300 y = 300 25 y = 12 Thus, the value of y is 12 at x = 0 . Hence, the first coordinate is ( 0 , 12 ) . Substitute x = 30 in the linear equation 10 x + 25 y = 300 . 10 ( 30 ) + 25 y = 300 300 + 25 y = 300 25 y = 300 − 300 y = 0 Hence, the second coordinate is ( 30 , 0 ) . Therefore, the coordinates are ( 0 , 12 ) and ( 30 , 0 ) . Use the test point to determine the half plane which is to be shaded. Let the test point be ( 0 , 0 ) . Now substitute the test point ( 0 , 0 ) in 10 x + 25 y ≥ 300 . 10 ( 0 ) + 25 ( 0 ) ≥ 300 0 ≥ 300 It is noticed that the inequality 10 x + 25 y ≥ 300 is false

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ISBN: 8220102020252

Author: RITCHEY

Publisher: PEARSON

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Chapter 3.3, Problem 25E

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To find: The number of units of plants and the number of animals gathered with a minimum number of hours of labour.

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Chapter 3.3, Problem 24E

Chapter 3.3, Problem 26E

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The number of units of plants and the number of animals gathered with a minimum number of hours of labour.

Solution Summary: The author explains how the linear programing problem can be stated as follows: x and y represent the number of units of plants and animals, respectively.

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To find: The number of units of plants and the number of animals gathered with a minimum number of hours of labour.

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