The objective function is z=5x+8y.
The corner points of the bounded feasible region are (0,0),(3,1),(2,4),(0,3),(5,0).
In Linear Programming Problems, the maximum or minimum value of the objective function occurs at one of the corner points of the feasible region.
Evaluating z=5x+8y at each corner point:
| Corner Point (x, y) | Calculation | Value of z |
|---|---|---|
| (0, 0) | z = 5(0) + 8(0) = 0 + 0 | 0 |
| (3, 1) | z = 5(3) + 8(1) = 15 + 8 | 23 |
| (2, 4) | z = 5(2) + 8(4) = 10 + 32 | 42 |
| (0, 3) | z = 5(0) + 8(3) = 0 + 24 | 24 |
| (5, 0) | z = 5(5) + 8(0) = 25 + 0 | 25 |
Comparing all values:
At (0,0): z=0
At (3,1): z=23
At (2,4): z=42
At (0,3): z=24
At (5,0): z=25
The maximum value is 42, which occurs at the corner point (2,4).
Therefore, the maximum value of the objective function is 42.