Question

Mr. Smith decides to feed his pet Doberman pinscher a combination of two dog foods. Each can of brand A contains 4 units of​ protein, 1 unit of​ carbohydrates, and 2 units of fat and costs 80 cents. Each can of brand B contains 1 unit of​ protein, 1 unit of​ carbohydrates, and 4 units of fat and costs 50 cents. Mr. Smith feels that each day his dog should have at least 7 units of​ protein, 4 units of​ carbohydrates, and 12 units of fat. How many cans of each dog food should he give to his dog each day to provide the minimum requirements at the least​ cost?

Answer 1

Answer:

  3 cans of Brand B and 1 can of Brand A will minimize the daily cost

Step-by-step explanation:

Approach

Such a minimization problem can be approached a number of different ways:

• use the solver function of a spreadsheet program
• use an on-line Simplex Method calculator
• graph it

Problem Formulation

The problem is succinctly stated as ...

  Minimize 0.8x +0.5y subject to 4x+y≥7, x+y≥4, 2x+4y≥12.

Additional implicit constraints are: x, y ≥ 0.

These constraints come from units of protein, carbohydrate, and fat, respectively.

Solution

Shown in the first two attachments are the solution using an on-line Simplex Method calculator. For that, and for the graphical solution in the third attachment, we use x and y to represent the numbers of cans of Brand A and Brand B, respectively.

The calculator says the optimal solution is ...

• 1 can of Brand A
• 3 cans of Brand B

__

Since there are only two brand choices, we can use a 2-dimensional graph to display the inequalities. The feasible region is the area of overlap of the solutions to the three inequalities. The cost function is evaluated at each of the vertices of that region in order to find the one with minimum cost.

It can be convenient to graph the cost function as a line f(x, y) = constant, so that the slope of it can be compared to the boundaries of the feasible region. It is usually obvious visually which vertex will minimize the cost (put the cost line closest to the origin). In the attachment, the cost function is shown as a dashed line to differentiate it from the other boundary lines on the graph.

As with the on-line calculator, the graphical solution finds the optimum combination to be ...

  (brand A, brand B) = (1 can, 3 cans)