Suppose Bill is on a low-carbohydrate diet. He can eat only three foods: Rice Krispies, cottage cheese, and popcorn. The marginal utilities for each food are tabulated below. Bill is allowed only 167 grams of carbohydrates daily. Rice Krispies, cottage cheese, and popcorn provide 25, 6, and 10 grams of carbohydrates per cup, respectively. Referring to the accompanying table, respond to the following questions:
Units of Food (cups/day) Marginal utility of Rice Krispies Marginal utility of cottage cheese Marginal utility of popcorn
1 175 72 90
2 150 66 80
3 125 60 70
4 100 54 60
5 75 48 50
6 50 36 40
7 25 30 30
8 25 18 20
Given that Bill can consume only 167 grams of carbohydrates daily, how many cups of each food will he consume daily? Show your work.
To determine the number of cups of each food Bill will consume daily while staying within the 167 grams of carbohydrates limit, we need to consider the carbohydrate content and the marginal utilities of each food.
First, let's calculate the total carbohydrate content of each food item in relation to the number of cups consumed:
Rice Krispies: 25 grams of carbohydrates per cup
Cottage cheese: 6 grams of carbohydrates per cup
Popcorn: 10 grams of carbohydrates per cup
Now, let's set up the problem mathematically:
Let's say Bill consumes x cups of Rice Krispies, y cups of cottage cheese, and z cups of popcorn daily.
To stay within the daily limit of 167 grams of carbohydrates, we can set up the equation:
25x + 6y + 10z ≤ 167
Next, we need to maximize Bill's total utility, which involves comparing the marginal utilities of each food item.
By looking at the table, we can see that initially, Rice Krispies have the highest marginal utility but the least number of carbohydrates. As Bill consumes more Rice Krispies, its marginal utility decreases.
On the other hand, popcorn has the lowest marginal utility but provides the highest number of carbohydrates. As Bill consumes more popcorn, its marginal utility also decreases.
Cottage cheese falls in between, with a moderate marginal utility and carbohydrate content.
To find Bill's optimal consumption levels, we need to find the combination of x, y, and z that maximizes his total utility while still staying within the carbohydrate limit.
We can use a method like trial and error or linear programming optimization to solve this. Given that the table is not continuous, trial and error might be more appropriate.
By testing different values for x, y, and z while keeping within the carbohydrate limit, we can determine the combination that yields the highest total utility.
To determine how many cups of each food Bill will consume daily given that he can consume only 167 grams of carbohydrates, we need to calculate the total carbohydrates provided by each food and find the combination that adds up to 167 grams or less.
Here are the carbohydrate values per cup for each food:
- Rice Krispies: 25 grams/cup
- Cottage cheese: 6 grams/cup
- Popcorn: 10 grams/cup
To find the number of cups of each food Bill will consume daily, we will start from the highest marginal utility and gradually reduce it until we reach the maximum carbohydrate limit.
1. Start with 8 cups of Rice Krispies. This provides 8 x 25 = 200 grams of carbohydrates. Since this exceeds the limit of 167 grams, we can reduce the number of cups.
2. Reduce the number of Rice Krispies cups to 7. This provides 7 x 25 = 175 grams of carbohydrates.
3. Now, let's check the combination of Rice Krispies and cottage cheese. We have 25 grams of carbohydrates left to reach the limit of 167 grams. Since cottage cheese provides 6 grams of carbohydrates per cup, the maximum number of cups of cottage cheese we can add is 25 / 6 = 4 cups.
4. With 7 cups of Rice Krispies and 4 cups of cottage cheese, the total carbohydrates consumed is 7 x 25 + 4 x 6 = 175 + 24 = 199 grams. This exceeds the limit.
5. Therefore, we need to reduce the number of cups of Rice Krispies further. Let's try 6 cups. This gives us 6 x 25 = 150 grams of carbohydrates.
6. We have 167 - 150 = 17 grams of carbohydrates remaining to reach the limit. Since popcorn provides 10 grams of carbohydrates per cup, we can add a maximum of 17 / 10 = 1.7 cups. However, we cannot consume a fraction of a cup, so we consider only 1 cup.
7. With 6 cups of Rice Krispies, 1 cup of popcorn, and 4 cups of cottage cheese, the total carbohydrates consumed is 6 x 25 + 1 x 10 + 4 x 6 = 150 + 10 + 24 = 184 grams. This is within the limit.
Therefore, Bill will consume 6 cups of Rice Krispies, 1 cup of popcorn, and 4 cups of cottage cheese daily to stay within the 167 grams of carbohydrates limit.