Question

The question is as follows: "A diet is to contain at least 1144 units of carbohydrates, 1888 units of proteins, and 2196 calories. Two foods are available: F_{7} which costs $0.05 per unit and F₂ , which costs $0.08 per unit. Each unit of F_{7} contains 35 units of carbohydrates, 56 units of proteins, and 65 calories. Each unit of F₂ contains 50 units of carbohydrates, 20 units of proteins, and 45 calories. How many units of each food should be purchased to satisfy the dietary requirements at minimum cost? What is the minimum cost?" Please proceed with the calculations to determine the solution.

Answer 1

To find the minimum cost while meeting dietary requirements using foods F7 and F2, set up a linear programming problem with the objective function to minimize cost and constraints based on the dietary needs. Graph the inequalities or use a method like simplex to determine the optimal quantities of each food to purchase, then calculate the cost using these quantities.

Step-by-step explanation:

To determine how many units of each food, F7 and F2, should be purchased to satisfy the dietary requirements at minimum cost, we should set up a linear programming problem. Let x represent the number of units of F7 and y represent the number of units of F2. The objective function to minimize is the total cost: C = 0.05x + 0.08y.

The dietary requirements give us the following constraints:

• For carbohydrates: 35x + 50y ≥ 1144,
• For proteins: 56x + 20y ≥ 1888,
• For calories: 65x + 45y ≥ 2196.
• Additionally, x ≥ 0 and y ≥ 0 since we cannot purchase negative units of food.

By graphing these inequalities or using linear programming techniques like the simplex method, we can find the values of x and y that minimize C while still meeting the constraints. Once the optimal values for x and y are found, the minimum cost is computed using the objective function C.