Q.77 - Precalculus Enhanced with Graphing Utilities

Question

A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is chicken, corn, and 2% milk. Each serving of chicken has 30 g of protein, 35 g of carbohydrates, and 200 mg of calcium. Each serving of corn has 3 g of protein, 16 g of carbohydrates, and 10 mg of calcium. Each glass of 2% milk has 9 g of protein, 13 g of carbohydrates, and 300 mg of calcium. How many servings of each food should the dietitian provide for the patient?

Step-by-Step Solution

Answer

it is necessary to provide 1.5 chicken, 1 corn, 2 milk

Step 1: Given information

We are given that A dietitian wishes a patient to have a meal that has 66 grams (g) of protein, 94.5 g of carbohydrates, and 910 milligrams (mg) of calcium. 

Step 2: Find the equations

Let x be amount of protine in chicken y be protein in corn and z be the amount of protein in 2% milk

We get,

$$\begin{gathered} 30x+3y+9z=66 \left(1\right) \\ 35x+16y+13z=94.5 \left(2\right) \\ 200x+10y+300z=910 \left(3\right) \end{gathered}$$

Simplify the equation

$$\begin{gathered} 10x+y+3z=22 \left(4\right) \\ 35x+16y+13z=94.5 \left(5\right) \\ 20x+y+30z=91 \left(6\right) \end{gathered}$$

Step 3: Solve the equation

To solve the equation multiply equation 1 by -16

And add it to equation 2 

We get,

$-160x-16y-48z=-352+35x+16y+13z=94.5-125x-35z=-257.5$

We get, $125x-35z=-257.5$

Subtract equation 1 and 3

$10x+y+3z=22-20x-y-30z=-91-10x-27z=-69$

Multiply the first equation of the system by 2 and the second by 25 and add

$-250x-70z=-515+250x+675z=1725605z=1210$

Hence z=2

We find the value of x, ,y

$$\begin{gathered} 10x+27z=69 \\ 10x+54=69 \\ 10x=15 \\ x=1.5 \end{gathered}$$

Now 

$$\begin{gathered} -10x-y-3z=-22 \\ -15-y-6=-22 \\ -y=-1 \\ y=1 \end{gathered}$$

Step 4: Conclusion

it is necessary to provide 1.5 chicken, 1 corn, 2 milk