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8 July, 11:11

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.)

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  1. 8 July, 11:43
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    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
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