The manager of the school cafeteria is planning a plate lunch. She can spend no more than $2.00 per lunch and can
choose servings from selections A and B. The table indicates the cost and the quantity of sugar and protein (in grams)
per serving of each food choice.
Food Cost per serving Sugar Protein
A 20 cents 7 grams 6 grams
B 40 cents 3 grams 9 grams
It is recommended that the lunch contain at most 30 grams of sugar and at least 50 grams of protein. Note: Fractional
servings of each of the food choices can be prepared.
1. Is it possible to prepare a lunch that contains four servings of Food A and three servings of Food B and still satisfy the
constraints on cost, amount of sugar, and amount of protein? Explain.
2. Let a represent the number of servings of food A and let b represent the number of servings of food B. Write a set of
inequalities that model the constraints on cost, amount of sugar, and amount of protein.
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1. To determine if it is possible to prepare a lunch with four servings of Food A and three servings of Food B while satisfying the constraints, we need to check if the cost, sugar, and protein requirements are met.
Cost:
Four servings of Food A cost 4 * $0.20 = $0.80.
Three servings of Food B cost 3 * $0.40 = $1.20.
The total cost is $0.80 + $1.20 = $2.00, which is within the maximum limit of $2.00 per lunch.
Sugar:
Four servings of Food A contain 4 * 7 grams = 28 grams of sugar.
Three servings of Food B contain 3 * 3 grams = 9 grams of sugar.
The total sugar is 28 grams + 9 grams = 37 grams, which exceeds the maximum limit of 30 grams of sugar for the lunch. Therefore, this combination does not satisfy the sugar constraint.
Protein:
Four servings of Food A contain 4 * 6 grams = 24 grams of protein.
Three servings of Food B contain 3 * 9 grams = 27 grams of protein.
The total protein is 24 grams + 27 grams = 51 grams, which meets the minimum requirement of at least 50 grams of protein for the lunch.
In conclusion, a lunch with four servings of Food A and three servings of Food B does not satisfy the sugar constraint, as the total sugar exceeds the maximum limit. Therefore, it is not possible to prepare this lunch while meeting all the constraints.
2. Let's write a set of inequalities to model the constraints on cost, sugar, and protein.
Let a = number of servings of Food A.
Let b = number of servings of Food B.
Cost Constraint:
The cost per lunch should be at most $2.00.
The cost of a serving of Food A is $0.20 and the cost of a serving of Food B is $0.40.
The cost constraint can be written as: 0.20a + 0.40b ≤ 2.00.
Sugar Constraint:
The lunch should contain at most 30 grams of sugar.
Each serving of Food A contains 7 grams of sugar, and each serving of Food B contains 3 grams of sugar.
The sugar constraint can be written as: 7a + 3b ≤ 30.
Protein Constraint:
The lunch should contain at least 50 grams of protein.
Each serving of Food A contains 6 grams of protein, and each serving of Food B contains 9 grams of protein.
The protein constraint can be written as: 6a + 9b ≥ 50.
In summary, the set of inequalities that model the constraints are:
0.20a + 0.40b ≤ 2.00 (Cost constraint)
7a + 3b ≤ 30 (Sugar constraint)
6a + 9b ≥ 50 (Protein constraint)