Answer:
Explanation:
To solve the maximization problem graphically, we first need to graph the constraints and find the feasible region. Then we can find the corner points of the feasible region and evaluate the objective function at each corner point to find the maximum value.
Graphing the constraint 2x + 2y ≥ 10, we get the following line:
2x + 2y = 10
y = 5 - x
Graphing the constraint 3x + 6y ≤ 54, we get the following line:
3x + 6y = 54
y = 9 - 0.5x
We also need to graph the non-negative constraints x ≥ 0 and y ≥ 0, as well as the constraints x ≤ 10 and y ≤ 8, which limit the values of x and y.
The feasible region is the shaded region in the graph. To find the corner points of the feasible region, we can find the points where the lines intersect. There are four corner points: (0, 5), (0, 9), (6, 4), and (10, 0).
Now we can evaluate the objective function at each corner point to find the maximum value:
P(0, 5) = 82(0) + 50(5) = 250
P(0, 9) = 82(0) + 50(9) = 450
P(6, 4) = 82(6) + 50(4) = 652
P(10, 0) = 82(10) + 50(0) = 820
Therefore, the maximum value of the objective function is 820, which occurs at the point (10, 0).
Therefore, the maximum value of P(x,y)=82x+50y subject to the given constraints is 820, and it occurs at the point (10,0).