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,what is the number of pounds of kidney beans included in the diet?
A)1
B)2
C)3
D)4

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,what is the number of pounds of kidney beans included in the diet?
A)1
B)2
C)3
D)4

Quizplus · Accepted Answer

Let x be the number of pounds of kidney beans and y be the number of pounds of tofu.
The objective is to minimize the cost, which is given by:
Cost = 0.9x + 1.5y
Subject to the following constraints:
15x + 6y ≥ 30 (minimum calcium requirement)
10x + 10y ≥ 40 (minimum protein requirement)
3x + 9y ≥ 18 (minimum calorie requirement)
x ≥ 0 and y ≥ 0 (non-negativity constraint)
Using a graphing calculator or linear programming software, we can find that the optimum solution is x=3 and y=1.2.
Since the school can only buy whole pounds of kidney beans and tofu, the number of pounds of kidney beans in the diet is rounded up to the nearest integer, which is 3.
Therefore, the answer is C)3.

A School Is Trying to Determine a Nutritional Diet to Feed