Explanation Let x represent the number of units of Food C and y represent the number of units of Food D. Use the given information to prepare the following table. Food C Food D Total Number of Units x y Grams of Protein 5 10 30 Grams of Fat 5 5 50 Contract y 1 cost 20 28 The shelters have at least 30 gram of protein. Thus, from the above table the protein constraints is 5 x + 10 y ≥ 30 . The shelter have at least 20 gram of fat per feeding period. Thus, from the above table the fat constraints is 5 x + 5 y ≥ 20 . The contract require that at least 1 unit of D it can be represented as y ≥ 1 . The cost per serving from the above table, it can be represented as z = 20 x + 28 y . The linear program problem can be stated as follows. Minimize z = 20 x + 28 y . 5 x + 10 y ≥ 30 (1) 5 x + 5 y ≥ 20 (2) y ≥ 1 (3) 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 inequality symbol in the given equation 5 x + 10 y ≥ 30 with “equal to” symbol. Then, the equation (1) becomes, 5 x + 10 y = 30 . Obtain any 2 points of the above equation to plot on the line. Consider x = 0 , 6 and obtain the corresponding values of y respectively. Substitute x = 0 in the linear equation 5 x + 10 y = 30 . 5 ( 0 ) + 10 y = 30 10 y = 30 y = 30 10 y = 3 Thus, the value of y is 3 at x = 0 . Hence, the first coordinate is ( 0 , 3 ) . Substitute x = 6 in the linear equation 5 x + 10 y = 30 . 5 ( 6 ) + 10 y = 30 30 + 10 y = 30 10 y = 30 − 30 y = 0 Thus, the value of y is 0 at x = 6 . Hence, the second coordinate is ( 6 , 0 ) . Therefore, the coordinates are ( 0 , 3 ) and ( 6 , 0 ) . Now determine whether the inequality is true or false using test point. Let the test point be ( 0 , 0 ) . Now substitute the test point ( 0 , 0 ) in 5 x + 10 y ≥ 30 . 5 ( 0 ) + 10 ( 0 ) ≥ 30 0 ≥ 30 It is noticed that, the inequality 5 x + 10 y ≥ 30 is false. Thus, shade the half-plane that does not contain the test point ( 0 , 0 ) . 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 inequality symbol in the given equation 5 x + 5 y ≥ 20 with “equal to” symbol. Then, the equation (2) becomes, 5 x + 5 y = 20 . Obtain any 2 points of the above equation to plot on the line. Consider x = 0 , 4 and obtain the corresponding values of y respectively. Substitute x = 0 in the linear equation 5 x + 5 y = 20

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 3.3, Problem 3YT

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The amount of each food must be bought to produce the minimum cost per serving.

Solution Summary: The author explains that the minimum cost of each food is 96 cents per serving. The protein constraints are 5x+10yge 30.

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Chapter 3 Solutions