Question

The maximum value of $ P = 500x + 400y $ for the given constraints $ x + y \leq 200, \quad x \geq 20, \quad y \geq 4x, \quad y \geq 0 $

• A. 96,000
• B. 84,000
• C. 98,000
• D. 82,000

Hint:

In linear programming, the maximum or minimum value of the objective function occurs at one of the corner points of the feasible region. Make sure to evaluate the objective function at all corner points to find the optimal value.

Answer 1

The Correct Option is B

We are given the following linear programming problem: \[ P = 500x + 400y \] subject to the constraints: \[ x + y \leq 200 \quad \text{(Constraint 1)} \] \[ x \geq 20 \quad \text{(Constraint 2)} \] \[ y \geq 4x \quad \text{(Constraint 3)} \] \[ y \geq 0 \quad \text{(Constraint 4)} \] We need to maximize \( P \) subject to these constraints.
Step 1: Plot the constraints
We begin by plotting the constraints to find the feasible region. 1. From \( x + y \leq 200 \), rearranging gives: \[ y = 200 - x \] This is a straight line with a slope of -1. 2. From \( x \geq 20 \), we have a vertical line at \( x = 20 \). 3. From \( y \geq 4x \), we have a line with a slope of 4. 4. From \( y \geq 0 \), this corresponds to the x-axis.
Step 2: Find the corner points
To solve the linear programming problem, we need to determine the corner points of the feasible region by solving the system of equations formed by the intersection of the constraints. - Intersection of \( x + y = 200 \) and \( y = 4x \): \[ x + 4x = 200 \quad \Rightarrow \quad 5x = 200 \quad \Rightarrow \quad x = 40 \] Substituting \( x = 40 \) into \( y = 4x \), we get \( y = 160 \). So, one point is \( (40, 160) \). - Intersection of \( x = 20 \) and \( y = 4x \): \[ y = 4(20) = 80 \] So, another point is \( (20, 80) \). - Intersection of \( x = 20 \) and \( x + y = 200 \): \[ 20 + y = 200 \quad \Rightarrow \quad y = 180 \] So, another point is \( (20, 180) \).
Step 3: Evaluate \( P \) at the corner points
Now, we evaluate the objective function \( P = 500x + 400y \) at each of the corner points. - At \( (40, 160) \): \[ P = 500(40) + 400(160) = 20,000 + 64,000 = 84,000 \] - At \( (20, 80) \): \[ P = 500(20) + 400(80) = 10,000 + 32,000 = 42,000 \] - At \( (20, 180) \): \[ P = 500(20) + 400(180) = 10,000 + 72,000 = 82,000 \]
Step 4: Conclusion
The maximum value of \( P \) occurs at the point \( (40, 160) \), and the maximum value of \( P \) is 84,000. Thus, the correct answer is \( 84,000 \).