To find the maximum value of the objective function $Z$, we need to evaluate $Z$ at each corner point (vertex) of the feasible region. The problem provides the coordinates for points O, A, C, and B in the yellow cells on the right side of the spreadsheet.
1. Identify the Objective Function:
From cell C2, the formula for $Z$ is:
$$Z = 250X + 75Y$$
2. Evaluate Z at each Corner Point:
*
Point O $(0, 0)$:
$$Z = 250(0) + 75(0) = 0$$
*(This matches the value in cell AH8)*
*
Point A $(0, 60)$:
$$Z = 250(0) + 75(60)$$
$$Z = 0 + 4500 = 4500$$
*(This matches the value in cell AH13)*
*
Point C $(20, 0)$:
$$Z = 250(20) + 75(0)$$
$$Z = 5000 + 0 = 5000$$
*(This matches the value in cell AH18)*
*
Point B $(10, 50)$:
$$Z = 250(10) + 75(50)$$
$$Z = 2500 + 3750$$
$$Z = 6250$$
*(This matches the calculated value shown in cell AH24)*
3. Determine the Maximum Value:
Comparing the values of $Z$ at all corners:
- $O: 0$
- $A: 4500$
- $C: 5000$
- $B: 6250$
The highest value is 6250, which occurs at Point B.
Final Answer:
The maximum value of Z is 6250.
Parent Tip: Review the logic above to help your child master the concept of optimization with linear programming worksheet.
