Q. 78 - Precalculus Enhanced with Graphing Utilities

Question

Nutrition A dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, 58 g of carbohydrates, and 630 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and 2% milk. Each serving of pork chops has 23 g of protein, 0 g of carbohydrates, and 10 mg of calcium. Each serving of corn on the cob contains 3 g of protein, 16 g of carbohydrates, and 10 mg of calcium. Each glass of 2% milk contains 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

The dietician should give the patient 1 pork chops, 2 corn on the cab and 2.2 %. 

Step 1. Given information

We have been given that a dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, 58 g of carbohydrates, and 630 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and 2% milk. Each serving of pork chops has 23 g of protein, 0 g of carbohydrates, and 10 mg of calcium. Each serving of corn on the cob contains 3 g of protein, 16 g of carbohydrates, and 10 mg of calcium. Each glass of 2% milk contains 9 g of protein, 13 g of carbohydrates, and 300 mg of calcium. 

We have to find how many servings of each food should the dietitian provide for the patient.

Step 2. Make a table to understand the given conditions

Total amount	Protein (47 g)	Carbohydrates (58 g)	Calcium (630 mg)
x	23	0	10
y	3	16	10
z	9	13	300

Step 3. Write the system of equations and corresponding augmented matrix

$\begin{array}{l}23x+3y+9z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=47\\ 16y+13z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=58\\ 10x+10y+300z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=630\end{array}$

The corresponding augmented matrix is:

$\left[\begin{array}{cccc}23& 3& 9& 47\\ 0& 16& 13& 58\\ 10& 10& 300& 630\end{array}\right]$

Step 4. Perform elementary row operations

$$\begin{gathered} R_3\to \frac{1}{10}{R}_3: \\ \left[\begin{array}{cccc}23& 3& 9& 47\\ 0& 16& 13& 58\\ 1& 1& 30& 63\end{array}\right] \\ {R}_1\leftrightarrow R_3 \\ \left[\begin{array}{cccc}1& 1& 30& 63\\ 0& 16& 13& 58\\ 23& 3& 9& 47\end{array}\right] \\ {R}_3\to R_3-23R_1 \\ \left[\begin{array}{cccc}1& 1& 30& 63\\ 0& 16& 13& 58\\ 0& -20& -681& -1402\end{array}\right] \end{gathered}$$

Step 5. Perform more row operations

$$\begin{gathered} \begin{array}{r}{R}_3\to R_3+\frac{20}{16}{R}_2:\end{array} \\ \left[\begin{array}{cccc}1& 1& 30& 63\\ 0& 16& 13& 58\\ 0& 0& -\frac{2659}{4}& -\frac{2659}{2}\end{array}\right] \\ \begin{array}{r}{R}_3\to -\frac{4}{2659}{R}_3\\ \left[\begin{array}{cccc}1& 1& 30& 63\\ 0& 16& 13& 58\\ 0& 0& 1& 2\end{array}\right]\end{array} \end{gathered}$$

Step 6. Solve for x, y and z

$$\begin{gathered} \begin{array}{l}x+y+30z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=63\\ 16y+13z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=58\\ z\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=2\end{array} \\ z=2 \\ From \left(2\right):16y+13\cdot 2=58 \\ 16y=32 \\ y=2 \\ From \left(1\right):x+2+30\cdot 2=63 \\ x=1 \end{gathered}$$

The dietician should give the patient 1 pork chops, 2 corn on the cab and 2.2 %.