Question

A biologist wishes to feed laboratory rabbits a mixture of two types of foods. Type I contains 8 g of fat, 12 g of carbohydrate, and 2 g of protein per ounce. Type II contains 16 g of fat, 12 g of carbohydrate, and 1 g of protein per ounce. Type I costs $0.20 per ounce and type II costs $0.30 per ounce. The rabbits each receive a daily minimum of 32 g of fat, 48 g of carbohydrate, and 4 g of protein, but get no more than 5 oz of food per day. How many ounces of each food type should be fed to each rabbit daily to satisfy the dietary requirements at minimum cost? Type I___oz Type II___oz

Answer 1

To solve this problem, set up a system of equations based on the given information and solve for the values of x and y that satisfy all the requirements and minimize cost.

Step-by-step explanation:

To solve this problem, we need to set up a system of equations based on the given information. Let's assume that the rabbits are fed x ounces of Type I food and y ounces of Type II food.

The total fat requirement can be expressed as 8x + 16y ≥ 32 (since each ounce of Type I food contains 8 g of fat and each ounce of Type II food contains 16 g of fat).

The total carbohydrate requirement can be expressed as 12x + 12y ≥ 48 (since each ounce of both Type I and Type II food contains 12 g of carbohydrate).

The total protein requirement can be expressed as 2x + y ≥ 4 (since each ounce of Type I food contains 2 g of protein and each ounce of Type II food contains 1 g of protein).

We also need to consider the constraint of not exceeding 5 ounces of food per day, so we add the condition: x + y ≤ 5.

To minimize cost, we want to minimize the total cost, which can be expressed as 0.20x + 0.30y.

We can now solve this system of equations using a method such as substitution or elimination to find the values of x and y that satisfy all the requirements and minimize cost.