A dog food is to be mixed from three kinds of ingredients: rolled oats, beef parts, and whey. The cost of each is 20 cents, 40 cents, and 35 cents per lb. Each ingredient contains 50, 450, and 200 units of protein and 350, 65, and 100 units of carbohydrate respectively. The minimum daily requirements of protein and carbohydrate are 250 and 300 units respectively.
Find a minimum cost, to the nearest penny, of a mixture which satisfies the minimum daily requirements.
ALSO, How much does such a mixture weigh?
To find the minimum cost of the mixture that satisfies the minimum daily requirements, we can use linear programming.
Let's define the variables:
- Let x be the number of pounds of rolled oats in the mixture.
- Let y be the number of pounds of beef parts in the mixture.
- Let z be the number of pounds of whey in the mixture.
Now let's set up the equations based on the constraints and the objective function:
1. Constraints:
- Protein constraint: 50x + 450y + 200z ≥ 250
- Carbohydrate constraint: 350x + 65y + 100z ≥ 300
- Non-negativity constraint: x ≥ 0, y ≥ 0, z ≥ 0
2. Objective function:
- Cost: 0.20x + 0.40y + 0.35z
Now we can solve this linear programming problem using a suitable method or software (such as the simplex method or linear programming solver). The solution will yield the minimum cost and the values of x, y, and z.
Next, to find the weight of the mixture, we simply need to sum the weights of the individual ingredients. So the weight of the mixture would be x + y + z, where x, y, and z are the values we obtained from solving the linear programming problem.
Note: Linear programming problems can also be solved graphically by plotting the constraints and finding the feasible region. However, due to the complexity of this problem with multiple variables, it is more efficient to use a solver.