Final answer:
By using the method of corners for the linear programming problem, the optimal solution with the highest objective function value is found to be at the point (4, 7), which is not listed in the choices provided.
Step-by-step explanation:
To solve the linear programming problem by the method of corners, we need to find the feasible region and then evaluate the objective function at all the corner points. The constraints given are:
- 5x - y ≤ 37
- 5x + 5y ≤ 45
- y ≤ 7
- x ≥ 0
- y ≥ 0
We can start by graphing these inequalities to define the feasible region. Since this is a maximization problem, we want to find the corner of the feasible region that gives us the highest value for the objective function p = 5x + 8y.
The corner points are where the lines intersect, and we can find these by solving the system of equations:
- At x = 0, y = 0 which gives p = 5(0) + 8(0) = 0
- At x = 0, y = 7 (from the constraint y ≤ 7) which gives p = 5(0) + 8(7) = 56
- By solving 5x + 5y = 45 and y = 7, we get x = 4 and y = 7 which gives p = 5(4) + 8(7) = 76
- By solving 5x - y = 37 and y = 0, we get x = 37/5 and y = 0 which gives p = 5(37/5) + 8(0) = 37
- By solving 5x - y = 37 and 5x + 5y = 45, we get x = 3 and y = 4 which gives p = 5(3) + 8(4) = 47
The highest value of the objective function is at the point (4, 7), thus it is the optimal solution. So, option (c) (5, 2) is not the correct answer. The correct answer is not provided in the choices as the maximum value for p is achieved at the point (4, 7) with p = 76.