Final answer:
To solve the given LP problem using the corner point graphical method, we need to plot the feasible region represented by the system of inequalities and find the corner points. The maximum profit occurs at the corner point (10, 0) with a profit of $40. The slack for each constraint at the optimal solution is also calculated.
Step-by-step explanation:
To solve the given LP problem using the corner point graphical method, we need to plot the feasible region represented by the system of inequalities. The inequalities are:
- 3X + 5Y ≤ 150
- X - 2Y ≤ 10
- X + 3Y ≤ 15
- X ≥ 0
- Y ≥ 0
Graphing these inequalities, we find that the feasible region is the triangle bounded by the lines 3X + 5Y = 150, X - 2Y = 10, and X + 3Y = 15. To solve the LP problem, we need to find the corner points of this feasible region.
Next, we calculate the objective function (profit) at each corner point. The corner points are (0, 30), (0, 10), and (10, 0). Calculating the profit at each corner point, we find that (10, 0) corresponds to the maximum profit of $40.
Finally, we calculate the slack for each constraint at the optimal solution. The slack for the first constraint, 3X + 5Y ≤ 150, is 150 - (3 * 10) - (5 * 0) = 120. The slack for the second constraint, X - 2Y ≤ 10, is 10 - (0 * 10) - (2 * 0) = 10. The slack for the third constraint, X + 3Y ≤ 15, is 15 - (10 * 1) - (3 * 0) = 5.