Give step-by-step solution with explanation and final answer:A dietician works in a hospital and prepares meals under the guidance of a physician. Suppose that for a particular patient, a physician pr

Question

Give step-by-step solution with explanation and final answer:A dietician works in a hospital and prepares meals under the guidance of a physician. Suppose that for a particular patient, a physician prescribes a meal to have 650
calories, 45 g of protein, and 340 mg of vitamin C. The dietician decides to prepare the meal using steak (each 3-02 serving contains 300 Cal, 20 g of protein, and no
vitamin C), baked potatoes (one baked potato contains 100 Cal, § g of protein, and 20 mg of vitamin C), and Brussels sprouts (one 156-g serving contains 50 Cal, 5 g
of protein, and 100 mg of vitamin C). How many servings of each food are required to satisfy the physician's requirements?

How many servings of steak are required? |_|

How many baked potatoes are required? |

How many servings of Brussels sprouts are required? [|

VIPSolutions · Accepted Answer

Let's translate the word problem into a system of equations.

**Let:**
- $ x $ = number of servings of steak
- $ y $ = number of servings of baked potatoes
- $ z $ = number of servings of Brussels sprouts

**Nutritional Values:**
| Food             | Calories | Protein (g) | Vitamin C (mg) |
|------------------|----------|-------------|----------------|
| 1 Steak Serving  | 300      | 20          | 0              |
| 1 Potato         | 100      | 5           | 20             |
| 1 Brussels Sprt. | 50       | 5           | 100            |

Physician's requirements:
- **Calories:** 650
- **Protein:** 45 g
- **Vitamin C:** 340 mg

### 1. Set up equations

#### Calories:
$ 300x + 100y + 50z = 650 $

#### Protein:
$ 20x + 5y + 5z = 45 $

#### Vitamin C:
$ 0x + 20y + 100z = 340 $

---

### 2. Solve the system step-by-step

#### From Vitamin C equation:
$ 20y + 100z = 340 $  
Divide by 20:  
$ y + 5z = 17 $  
So,  
$ y = 17 - 5z $

---

#### Substitute y in the protein equation:
$ 20x + 5y + 5z = 45 $  
$ 20x + 5(17 - 5z) + 5z = 45 $  
$ 20x + 85 - 25z + 5z = 45 $  
$ 20x + 85 - 20z = 45 $  
$ 20x - 20z = 45 - 85 $  
$ 20x - 20z = -40 $  
$ x - z = -2 $  
So,  
$ x = z - 2 $

---

#### Substitute x and y in the calories equation:
$ 300x + 100y + 50z = 650 $  
Replace $ x $ and $ y $:  
$ 300(z - 2) + 100(17 - 5z) + 50z = 650 $  
$ 300z - 600 + 1700 - 500z + 50z = 650 $  
$ (300z - 500z + 50z) + (-600 + 1700) = 650 $  
$ (-150z) + 1100 = 650 $  
$ -150z = 650 - 1100 $  
$ -150z = -450 $  
$ z = \frac{-450}{-150} = 3 $

---

### 3. Back substitute to find x and y

#### $ z = 3 $

$ x = z - 2 = 3 - 2 = 1 $

$ y = 17 - 5z = 17 - 5 \times 3 = 17 - 15 = 2 $

---

## Final Answers

**How many servings of steak are required?**
$$
\boxed{1}
$$

**How many baked potatoes are required?**
$$
\boxed{2}
$$

**How many servings of Brussels sprouts are required?**
$$
\boxed{3}
$$

---

### **Check the answer:**

- Calories: $1 \times 300 + 2 \times 100 + 3 \times 50 = 300 + 200 + 150 = 650$
- Protein: $1 \times 20 + 2 \times 5 + 3 \times 5 = 20 + 10 + 15 = 45$
- Vitamin C: $2 \times 20 + 3 \times 100 = 40 + 300 = 340$

All requirements are met.

**Final Answer:**
- Steak: $1$ serving
- Baked potatoes: $2$ servings
- Brussels sprouts: $3$ servings

VIPSolutions · Answer

To determine the number of servings needed, we set up a system of linear equations based on the nutritional content of each food item and the patient's requirements. By solving these equations simultaneously, we find the exact quantities of each food item that satisfy all nutritional constraints.

✨ VIPSolutions

Question:

Final Answer:

Question:

Final Answer:

📢 Instructions:

Study Help, Fast Answers

✨ VIPSolutions

Question:

Final Answer:

Question:

Final Answer:

📢 Instructions:

Related Answers

Study Help, Fast Answers