Chapter 11.1, Problem 77AYU

To determine

To calculate: The amount of serving of chicken, corn and $2\%$ milk that the dietician should provide for the patient.

Answer to Problem 77AYU

The amount of serving of chickenis $\frac{3}{2}$ , corn is $1$ and $2\%$ milkis 2 that the dietician should provide for the patient.

Explanation of Solution

Given information:

A patient should be given 66 grams of protein, 94.5 grams carbohydrate and 910 milligrams. The food service of hospital include chicken, corn and $2\%$ milk. 30 grams of protein, 35 grams of carbohydrates and 200 milligrams of calcium is present in chicken. 3 grams of protein, 16 grams of carbohydrates and 10 milligrams of calcium is present in corn. 9 grams of protein, 13 grams of carbohydrates and 300 milligrams of calcium is present in $2\%$ milk.

Calculation:

Consider the provided statement that a patient should be given 66 grams of protein, 94.5 grams carbohydrate and 910 milligrams. The food service of hospital include chicken, corn and $2\%$ milk. 30 grams of protein, 35 grams of carbohydrates and 200 milligrams of calcium is present in chicken. 3 grams of protein, 16 grams of carbohydrates and 10 milligrams of calcium is present in corn. 9 grams of protein, 13 grams of carbohydrates and 300 milligrams of calcium is present in $2\%$ milk.

Let the patient is given x amount of chicken, y amount of corn and z amount of $2\%$ milk.

Consider the system of equations formulated from the provided data,

  $\begin{array}{c}30x+3y+9z=66\\ 35x+16y+13z=94.5\\ 200x+10y+300z=910\end{array}$

Multiply first equation by 10 and third equation by 3,

  $\begin{array}{c}300x+30y+90z=660\\ 600x+30y+900z=2730\end{array}$

Use the method of elimination to solve the system above.

Subtract both the equations,

  $\begin{array}{c}-300x-810z=-2070\\ 300x+810z=2070\end{array}$

Multiply second equation by 5 and third equation by 8,

  $\begin{array}{c}175x+80y+65z=472.5\\ 1600x+80y+2400z=7280\end{array}$

Subtract both the equations,

  $1425x+2335z=6807.5$

Now, solve the equations $1425x+2335z=6807.5$ and $300x+810z=2070$ .

To obtain $x=\frac{3}{2},z=2$ .

Now substitute $x=\frac{3}{2},z=2$ in the equation $30x+3y+9z=66$ .

Therefore, $y=1$ .

Therefore, solution is the coordinate pair, $\left(\frac{3}{2},1,2\right)$ .

Thus, the amount of serving of chicken is $\frac{3}{2}$ , corn is $1$ and $2\%$ milkis 2 that the dietician should provide for the patient.