Q. 77 - Precalculus Enhanced with Graphing Utilities

Question

Nutrition A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams (g) of protein, 59 g of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has 30 g of protein, 20 g of carbohydrates, and 2 mg of vitamin A. Each serving of baked eggs contains 15 g of protein, 2 g of carbohydrates, and 20 mg of vitamin A. Each serving of acorn squash contains 3 g of protein, 25 g of carbohydrates, and 32 mg of vitamin A. How many servings of each food should the dietitian provide for the patient? 

Step-by-Step Solution

Answer

The dietician should give the patient 1.5 salmon steak, 2 baked eggs and 1 acorn squash.

Step 1. Given information

We have been given that Nutrition a dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams (g) of protein, 59 g of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has 30 g of protein, 20 g of carbohydrates, and 2 mg of vitamin A. Each serving of baked eggs contains 15 g of protein, 2 g of carbohydrates, and 20 mg of vitamin A. Each serving of acorn squash contains 3 g of protein, 25 g of carbohydrates, and 32 mg of vitamin A. 

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 (78 g)	Carbohydrates (59 g)	Vitamin A (75 mg)
x	30	20	2
y	15	2	20
z	3	25	32

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

$$\begin{gathered} \begin{array}{r}30x+15y+3z=78\\ 20x+2y+25z=59\\ 2x+20y+32z=75\end{array} \\ \mathrm{The} \mathrm{corresponding} \mathrm{augmented} \mathrm{matrix} \mathrm{is}: \\ \left[\begin{array}{cccc}30& 15& 3& 78\\ 20& 2& 25& 59\\ 2& 20& 32& 75\end{array}\right] \end{gathered}$$

Step 4. Perform elementary row operations

$$\begin{gathered} R_3\leftrightarrow R_1 \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 20& 2& 25& 59\\ 30& 15& 3& 78\end{array}\right] \\ {R}_2\to R_2-10R_1; \\ {R}_3\to R_3-15R_1 \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 0& -198& -295& -691\\ 0& -285& -477& -1047\end{array}\right] \end{gathered}$$

Step 5. Perform more row operations

$$\begin{gathered} \begin{array}{r}{R}_3\to R_3-\frac{285}{198}{R}_2\end{array} \\ :\left[\begin{array}{cccc}2& 20& 32& 75\\ 0& -198& -295& -691\\ 0& 0& -\frac{3457}{66}& -\frac{3457}{66}\end{array}\right] \\ {R}_3\to -\frac{66}{3457}{R}_3 \\ {R}_3\to R_3-\frac{285}{198}{R}_2: \\ \left[\begin{array}{cccc}2& 20& 32& 75\\ 0& -198& -295& -691\\ 0& 0& 1& 1\end{array}\right] \end{gathered}$$

Step 6. Solve for x, y and z

$$\begin{gathered} 2x+20y+32z=75 \left(1\right) \\ -198y-295z=-691 \left(2\right) \\ z=1 \left(3\right) \\ From \left(2\right):-198y-295=-691 \\ -198y=-396 \\ y=2 \\ From \left(1\right):2x+20\cdot 2+32\cdot 1=75 \\ 2x+72=75 \\ 2x=3 \\ x=1.5 \end{gathered}$$

The dietician should give the patient 1.5 salmon steak, 2 baked eggs and 1 acorn squash.