A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply zero units of protein, 2 units of carbohydrates, and 1 unit of fat, and will contain 18 calories. Each ounce of nuts will supply 3 units of protein, 1 unit of carbohydrate, and 2 units of fat, and will contain 28 calories. Every package must provide at least 9 units of protein, at least 8 units of carbohydrates, and no more than 9 units of fat. What is the least number of calories possible in a package? (Type an integer or a decimal.)

The least number of calories possible in a package is 129 calories.

Step-by-step explanation:

We can write equations to representeach requirement of the snack.

Fruit:0 protein

2 carb

1 fat

18 cal

Nuts: 3 prot

1 carb

2 fat

28

The amount of protein contained in the final snack will be: the sum of the amount of ounces of fruit (and nuts) times the amount of protein pero ounce of fruit (and nuts). So for N, M integers or decimals:

Protein=N*0 carbs+M*3 carbs, this means that fruit doesn't affect protein content.

Similarly we write the other requirements as inequalities:

a. Protein≥9 → N*0+M*3≥9

b. Carbs≥8 → N*2 + M*1≥8

c. Fat≤9 → N*1+M*2≤9

d. Cal=N*18+M*28

From a we get M≥3

Replacing M in b, N≥2.5.

And finally we replace in c to have a maximum amount.

2.5+2*3≤9

8.5<9 so this is the least amount of ounces in a package, avaluating how many calories each component contributes:

Least amount of calories: 18*2.5+28*3=129cal