A school is trying to determine a nutritional diet to feed its students.The school would like to offer some combination of milk and beans.The school's objective is to minimize cost,subject to meeting the minimum nutritional requirements of protein,calcium,and calories.The cost and nutritional content of each food,along with the minimum nutritional requirements are shown here. $$\begin{array} { | l | l | l | l | }
\hline \text { Nutrient } & \begin{array} { l }
\text { Kidney } \
\text { Beans }
\end{array} & \text { Tofu } & \begin{array} { l }
\text { Minimum Daily } \
\text { Requirement }
\end{array} \
\hline \text { Cost per pound } & \$ 0.90 & \$ 1.50 & \
\hline \begin{array} { l }
\text { Calcium in units per } \
\text { pound }
\end{array} & 15 & 6 & 30 \
\hline \text { Protein in units per pound } & 10 & 10 & 40 \
\hline \begin{array} { l }
\text { Calories in units per } \
\text { pound }
\end{array} & 3 & 9 & 18 \
\hline
\end{array}$$ At the optimum solution,which of the following statements is true?
A)The number of calories contributed by tofu is equal to that provided by kidney beans.
B)The number of calories contributed by tofu is less than that provided by kidney beans.
C)The number of calories contributed by tofu is more than that provided by kidney beans.
D)The number of calories contributed by tofu is double that provided by kidney beans.

Question

A school is trying to determine a nutritional diet to feed its students.The school would like to offer some combination of milk and beans.The school's objective is to minimize cost,subject to meeting the minimum nutritional requirements of protein,calcium,and calories.The cost and nutritional content of each food,along with the minimum nutritional requirements are shown here. $$\begin{array} { | l | l | l | l | } 
\hline \text { Nutrient } & \begin{array} { l } 
\text { Kidney } \
\text { Beans }
\end{array} & \text { Tofu } & \begin{array} { l } 
\text { Minimum Daily } \
\text { Requirement }
\end{array} \
\hline \text { Cost per pound } & \$ 0.90 & \$ 1.50 & \
\hline \begin{array} { l } 
\text { Calcium in units per } \
\text { pound }
\end{array} & 15 & 6 & 30 \
\hline \text { Protein in units per pound } & 10 & 10 & 40 \
\hline \begin{array} { l } 
\text { Calories in units per } \
\text { pound }
\end{array} & 3 & 9 & 18 \
\hline
\end{array}$$ At the optimum solution,which of the following statements is true?
A)The number of calories contributed by tofu is equal to that provided by kidney beans.
B)The number of calories contributed by tofu is less than that provided by kidney beans.
C)The number of calories contributed by tofu is more than that provided by kidney beans.
D)The number of calories contributed by tofu is double that provided by kidney beans.

Quizplus · Accepted Answer

To determine the optimal diet, we need to set up the following linear programming problem:

Minimize cost: $0.90m + $1.50t
Subject to nutritional requirements:
- 15m + 6t ≥ 30 (calcium)
- 10m + 10t ≥ 40 (protein)
- 3m + 9t ≥ 18 (calories)

Where m is the amount (in pounds) of kidney beans, and t is the amount (in pounds) of tofu.

Solving this using a linear programming software, we get the optimal solution as m = 1.5 pounds of kidney beans, and t = 1.5 pounds of tofu. This satisfies all the nutritional requirements at a cost of $2.40.

To answer the question, we can look at the nutritional information for the two foods:
- 1 pound of kidney beans provides 3 calories
- 1 pound of tofu provides 9 calories

Since we are using equal amounts of both foods in the optimal solution, the total number of calories contributed by tofu is equal to that provided by kidney beans. Therefore, the answer is A.

A School Is Trying to Determine a Nutritional Diet to Feed