Linear Programming?

Question

In the following linear program,

State what you are letting x and y be


Give the objective function


Write the constraints as linear inequalities


Show the graph. Be sure to do the shading.


List all corner points.


Write the answer in a sentence.


Cans of meat are to be mass produced and made available for distribution in emergency situations (such as earthquakes and floods). Each can of meat is to be a mixture of beef and pork and must contain at least 12 ounces of protein and 9 ounces of fat.


A pound of beef contains 5 ounces of protein and 3 ounces of fat while a pound of pork contains 3 ounces of protein and 3 ounces of fat.


If beef costs 50 cents a pound and pork costs 45 cents a pound, how many pounds of each should be used to make up a can of meat so that the nutritional requirements are met and the cost is minimized? What is the minimum cost?

Answer 1

Let X be pounds of beef in each can

Let Y be pounds of pork in each can

The amount of protein in the product must be at least 12 ounces:  5X + 3Y ≥ 12

The amount of fat in the product must be at least 9 ounces:  3X + 3Y ≥ 9

We also know that X ≥ 0  and Y ≥ 0

The objective function is:  cost = 0.50X + 0.45Y

Graph the inequalities, and shade above each (solid) equality line.

The intersection points will be (0, 4), (1.5, 1.5), and (3,0)

The point (1.5, 1.5) minimizes the objective function.

The amount of beef in the product should be 1.5 pounds, and the amount of pork in the product should be 1.5 pounds to minimize the cost of production.  This cost will be $1.43  (rounded up)