A dietecian is planning a snack package of fruits and nuts E

A dietecian is planning a snack package of fruits and nuts. Each ounce of fruit will supply zero units of protein, 3 units of carbohydrates, and 2 units of fat, and will contain 40 calories. Each ounce of nuts will supply 4 units of protein, 1 unit of carbohydrate, and 4 units of fat, and will contain 50 calories. Every package must provide at least 8 units of protein, at least 8 units of carbohydrates, and no more than 16 units of fat. Find the number of ounces of fruit and number of ounces nuts that will meet the requirement with the least number of calories. What is the least number of calories?

The dietician should use X ounce(s) of fruit and X ounce(s) of nuts. These amounts will have a total of X calories.

(Type whole numbers)

Solution

Let X1 be the ounce of fruit used and X2 be the ounce of nuts used.

Thus, total protien units is given by,

0X1 + 4X2 >= 8

total carbohydrate units is given by,

3X1 + 1X2 >= 8

total fat units is given by,

2X1 + 4X2 <= 16

Also, the total calories is given by,

Z = 40X1 + 50X2

So, the Linear programming problem is given by,

Minimize Z = 40X1 + 50X2

Subject to :

0X1 + 4X2 >= 8

3X1 + 1X2 >= 8

2X1 + 4X2 <= 16

X1, X2 >= 0

By solving the above linear programming problem using Excel, we get

X1 = 2, X2 = 2 and Z = 180.

Thus, the dietician should use 2 ounces of fruit and 2 ounces of nuts. These amounts will have a total of 180 calories.