Set up the objective function and the constraints, but do not solve.

Jack has a casserole and salad dinner. Each serving of casserole contains 270 calories, 5 milligrams of vitamins, and 8 grams of protein. Each serving of salad contains 30 calories, 6 milligrams of vitamins, and 1 gram of protein. Jack wants to consume at least 23 milligrams of vitamins and 29 grams of protein but keep the calories at a minimum. How many servings of each food should he eat? (Let x represent the number of servings of casserole, y the number of servings of salad, and C the number of calories.)

---Select---
 C =	
 	, subject to
vitamins    		
 
protein    		
 
x ≥ 0, y ≥ 0


We are tasked with setting up the **objective function** and the **constraints** for Jack's meal planning problem. The goal is to minimize Jack's caloric intake while ensuring he consumes at least the required amounts of vitamins and protein.

### Objective Function:
Jack wants to minimize the total number of calories, which comes from the servings of casserole and salad.

- Each serving of casserole provides 270 calories.
- Each serving of salad provides 30 calories.

Let \( x \) represent the number of servings of casserole and \( y \) represent the number of servings of salad. The total caloric intake \( C \) is:

\[
C = 270x + 30y
\]

### Constraints:
Jack has two nutritional requirements: vitamins and protein.

1. **Vitamins constraint**:
   - Each serving of casserole provides 5 milligrams of vitamins.
   - Each serving of salad provides 6 milligrams of vitamins.
   - Jack wants to consume at least 23 milligrams of vitamins.

\[
5x + 6y \geq 23
\]

2. **Protein constraint**:
   - Each serving of casserole provides 8 grams of protein.
   - Each serving of salad provides 1 gram of protein.
   - Jack wants to consume at least 29 grams of protein.

\[
8x + y \geq 29
\]

### Non-negativity constraint:
Jack can't consume a negative number of servings of casserole or salad, so:

\[
x \geq 0, \quad y \geq 0
\]

### Final Model:
Minimize:
\[
C = 270x + 30y
\]
subject to:
\[
5x + 6y \geq 23
\]
\[
8x + y \geq 29
\]
\[
x \geq 0, \quad y \geq 0
\]

Would you like to go over the problem details or constraints? Feel free to ask for clarification.

---

Here are five related questions:
1. How would changing the minimum vitamin requirement affect the objective function?
2. What happens if we increase the caloric intake per serving of salad?
3. How would adding a third food option change the model?
4. Can we solve this using the graphical method? How would the feasible region look?
5. What if Jack also wanted to maximize his protein intake while minimizing calories?

**Tip**: When setting up a linear programming problem, always identify the objective (minimize or maximize) and ensure the constraints reflect the real-world limitations clearly.

Jack has a casserole and salad dinner. Each serving of casserole contains 270 calories, 5 milligrams of vitamins, and 8 grams of protein. Each serving of salad contains 30 calories, 6 milligrams of vitamins, and 1 gram of protein. Jack wants to consume at least 23 milligrams of vitamins and 29 grams of protein but keep the calories at a minimum. How many servings of each food should he eat? (Let x represent the number of servings of casserole, y the number of servings of salad, and C the number of calories.)

Explore how to set up a linear programming model to minimize caloric intake for Jack's dinner while meeting his vitamin and protein requirements. This problem involves understanding objective functions, constraints, and inequalities, suitable for students in Grades 9-12.

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