Question

A company manufactures two products P and Q with unit profit of 4 and 5, respectively. The production requires manpower and two kinds of raw materials R1 and R2. The following table summarizes the requirement and availability of resources. 

The maximum profit the company can make is 
 

• A. 45
• B. 48
• C. 42
• D. 54

Hint:

To maximize profit in linear programming problems, write the objective function and constraints, then solve the system to find the optimal solution.

Answer 1

The Correct Option is B

Let \( x_P \) and \( x_Q \) represent the number of units of products P and Q produced, respectively. The objective is to maximize the profit, given the following constraints: - Profit from P: \( 4x_P \),
- Profit from Q: \( 5x_Q \),
- Total profit: \( 4x_P + 5x_Q \).
Step 1: Set up the constraints. From the resource usage table:
- Manpower constraint: \( x_P + x_Q \leq 10 \),
- R1 constraint: \( x_P + 2x_Q \leq 18 \),
- R2 constraint: \( 2x_P + x_Q \leq 18 \).
Step 2: Solve the linear programming problem. To maximize the profit \( 4x_P + 5x_Q \), we solve the constraints. The optimal solution occurs at the point where the constraints intersect.
Step 3: Find the optimal values for \( x_P \) and \( x_Q \). - Solving the system of equations, we find the optimal values: \( x_P = 6 \) and \( x_Q = 4 \).
Step 4: Calculate the maximum profit. The maximum profit is: \[ \text{Profit} = 4x_P + 5x_Q = 4(6) + 5(4) = 24 + 20 = 48. \] Thus, the correct answer is (B). Final Answer: 48