Billy Bob wants to gain some weight so that he can play football.Billy consumes only milk shakes and spinach.Milk shakes cost him $1 each and spinach costs $2 per serving.A milk shake has 850 calories and a serving of spinach has 200 calories.Billy Bob never spends more than $20 a day on food and he always consumes at least 8,000 calories per day.Which of the following is necessarily true?
A)Billy Bob consumes at least 9 milk shakes a day.
B)Billy Bob never consumes more than 6 servings of spinach a day.
C)Billy Bob never consumes positive amounts of both goods.
D)Billy Bob consumes only milk shakes.
E)None of the above.

Question

Quizplus · Accepted Answer

We know that Billy Bob never spends more than $20 a day on food. Let the number of milk shakes he consumes be M and the number of servings of spinach be S.

The cost of M milk shakes is $M, and the cost of S servings of spinach is $2S. So, we can write:

M + 2S ≤ 20

We also know that Billy Bob always consumes at least 8,000 calories per day. One milk shake has 850 calories, so M milk shakes would have 850M calories. One serving of spinach has 200 calories, so S servings of spinach would have 200S calories. So, we can write:

850M + 200S ≥ 8,000

To maximize his calorie intake, Billy Bob would want to consume as much of the cheaper item (milk shakes) as possible. The maximum number of milk shakes he can consume is 11, which would cost $11 and provide 9,350 calories. The maximum number of servings of spinach he can consume is 5, which would cost $10 and provide 1,000 calories. This satisfies both of the constraints above, and provides a total of 10,350 calories.

Therefore, Billy Bob never consumes more than 6 servings of spinach a day is necessarily true.

Billy Bob Wants to Gain Some Weight So That He