A dietitian wishes to plan a meal around three foods. The meal is to include 16,200 units of vitamin A, 4,620 units of vitamin C, and 680 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the table below. Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements.
Food I Food II Food III
Vitamin A 450 1800 1350
Vitamin C 120 530 385
Calcium 80 20 50
To determine the amount of each food the dietitian should include in the meal, we can set up a system of equations based on the given information.
Let's denote the amount of Food I, Food II, and Food III in ounces as x, y, and z, respectively.
From the table, we know that the following equations are true:
Total units of vitamin A: 450x + 1800y + 1350z = 16200
Total units of vitamin C: 120x + 530y + 385z = 4620
Total units of calcium: 80x + 20y + 50z = 680
We can now solve this system of equations to find the values of x, y, and z.
One way to solve this system is to use matrix algebra or a calculator with matrix capabilities to find the inverse of the coefficient matrix and then multiply it by the constant matrix. However, for a small system like this, we can use the method of substitution.
First, solve one equation for one variable in terms of the other two variables. Let's solve the third equation for x:
80x = 680 - 20y - 50z
x = (680 - 20y - 50z) / 80
Now substitute this expression for x in the first and second equations:
450((680 - 20y - 50z) / 80) + 1800y + 1350z = 16200
120((680 - 20y - 50z) / 80) + 530y + 385z = 4620
Simplify these equations:
3375 - 75y - 187.5z + 1800y + 1350z = 16200
816 - 30y - 75z + 530y + 385z = 4620
Combine like terms:
1125y + 1125z = 12825
500y + 310z = 3804
Now we have a system of two equations with two unknowns. We can solve this system using any method we prefer, such as substitution or elimination.
Let's solve this system using the method of substitution:
Solve the first equation for y:
y = (12825 - 1125z) / 1125
Substitute this expression for y in the second equation:
500((12825 - 1125z) / 1125) + 310z = 3804
Simplify this equation:
500(12825 - 1125z) + 310z = 3804(1125)
6406250 - 562500z + 310z = 4288500
Combine like terms:
-252190z = -2117750
Divide both sides by -252190:
z ≈ 8.4
Now we can substitute this value of z back into one of the equations to solve for y:
y = (12825 - 1125z) / 1125
y = (12825 - 1125(8.4)) / 1125
y ≈ 7.4
Finally, substitute the values of y and z back into one of the original equations to solve for x:
80x + 20y + 50z = 680
80x + 20(7.4) + 50(8.4) = 680
80x + 148 + 420 = 680
80x = 680 - 148 - 420
80x = 112
x = 112 / 80
x = 1.4
Therefore, the dietitian should include approximately 1.4 ounces of Food I, 7.4 ounces of Food II, and 8.4 ounces of Food III in the meal in order to meet the vitamin and calcium requirements.