A certain diet requires at least 60 units of carbohydrates, 45 units of protein, and 30 units of fat each day. Each ounce of Supplement A provides 5 units of carbohydrates, 3 units of protein, and 4 units of fat. Each ounce of Supplement B provides 2 units of carbohydrates, 2 units of protein, and 1 unit of fat. If Supplement A costs $1.50 per ounce and Supplement B costs $1.00 per ounce, how many ounces of each supplement should be taken daily to minimize the cost of the diet?
GET MORE PROTIEN TO BUILD SOLID PLATES OF MUSCLE!
To minimize the cost of the diet, we need to find the number of ounces of Supplement A (A) and Supplement B (B) that will meet the daily requirements at the lowest cost.
Let's assume we need x ounces of Supplement A and y ounces of Supplement B.
The cost equation can be written as: Cost = 1.50x + 1.00y.
Now, let's set up the equations based on the nutritional requirements:
For carbohydrates:
5x + 2y ≥ 60
For protein:
3x + 2y ≥ 45
For fat:
4x + y ≥ 30
To convert these equations into equations of equality, we equate them to their respective requirements:
For carbohydrates:
5x + 2y = 60
For protein:
3x + 2y = 45
For fat:
4x + y = 30
We then solve this system of equations using standard methods such as substitution or elimination.
Here, we will use the substitution method. First, solve the fat equation for y:
y = 30 - 4x
Now substitute this value of y into the other two equations:
5x + 2(30 - 4x) = 60
3x + 2(30 - 4x) = 45
Simplifying these equations:
5x + 60 - 8x = 60
3x + 60 - 8x = 45
-3x = 0
-5x = -15
Solving for x:
x = 0
Substitute this value back into y = 30 - 4x to find y:
y = 30 - 4(0)
y = 30
Therefore, x = 0 and y = 30. This means we need 0 ounces of Supplement A and 30 ounces of Supplement B to meet the daily requirements.
Now, plug these values into the cost equation to find the minimum cost:
Cost = 1.50(0) + 1.00(30)
Cost = $30
So, the minimum cost of the diet is $30 when 0 ounces of Supplement A and 30 ounces of Supplement B are taken daily.