CONSIDER THE FOLLOWING LINEAR PROGRAM:
Minimize C = 75P + 32Q
Subject To:
9P + 5Q >= 45        (1)
Q <= 6         (2)
P <=3           (3)
P>=0, Q>=0
If you place P in the horizontal axis, which of the following gives a value of 417 for the objective function?

(3, 6)

(1.67, 6)

(6, 3)

(3.60, 1.67)

To solve this problem, we need to calculate the value of the objective function \( C = 75P + 32Q \) for each of the given points \((P, Q)\) and check which one gives a value of 417.

### Objective Function:
\[ C = 75P + 32Q \]

### Given Points:

1. \((3, 6)\)
2. \((1.67, 6)\)
3. \((6, 3)\)
4. \((3.60, 1.67)\)

Now, let's calculate \( C \) for each point:

1. **For \((3, 6)\):**
   \[
   C = 75(3) + 32(6) = 225 + 192 = 417
   \]

2. **For \((1.67, 6)\):**
   \[
   C = 75(1.67) + 32(6) = 125.25 + 192 = 317.25
   \]

3. **For \((6, 3)\):**
   \[
   C = 75(6) + 32(3) = 450 + 96 = 546
   \]

4. **For \((3.60, 1.67)\):**
   \[
   C = 75(3.60) + 32(1.67) = 270 + 53.44 = 323.44
   \]

### Conclusion:
The point \((3, 6)\) gives a value of 417 for the objective function \( C \).

### Correct Answer:
\((3, 6)\)

Would you like more details on how the calculations were done or have any other questions?

Here are 5 related questions:
1. How would you graphically solve the linear programming problem?
2. What are the feasible regions for the given constraints?
3. How do you find the optimal solution in linear programming problems?
4. What happens if you change the coefficients in the objective function?
5. How do the constraints affect the feasible region in linear programming?

**Tip:** In linear programming, the optimal solution is often found at the vertices (corners) of the feasible region defined by the constraints.

CONSIDER THE FOLLOWING LINEAR PROGRAM: Minimize C = 75P + 32Q Subject To: 9P + 5Q >= 45 (1) Q <= 6 (2) P <= 3 (3) P >= 0, Q >= 0 If you place P in the horizontal axis, which of the following gives a value of 417 for the objective function? (3, 6), (1.67, 6), (6, 3), (3.60, 1.67)

Solve the linear programming problem where the objective is to minimize C = 75P + 32Q under the given constraints. Learn how to evaluate the objective function for multiple points and find the optimal solution. Suitable for Grades 10-12.

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