Final answer:
To find the maximum value of the objective function P=25x+45y given the constraints, we graph the system of inequalities and find the vertices of the feasible region. by calculating P for each vertex, we determine that the maximum value of P is 450 at the point (0,10).
Step-by-step explanation:
To determine the maximum value of the objective function, P, we must first graph the given system of inequalities. the constraints form a feasible region on a coordinate plane, within which we must find the point that maximizes the objective function P = 25x + 45y.
We graph the constraints:
- 4x + y ≤16 (Boundary line intersects x-axis at (4,0) and y-axis at (0,16))
- x + y ≤10 (Boundary line intersects x-axis at (10,0) and y-axis at (0,10))
- x ≥ 0 (x is non-negative)
- y ≥ 0 (y is non-negative)
The feasible region is a polygon formed by the intersection of these constraints. We evaluate the objective function P at each vertex of this polygon. the vertices of the feasible region are found by solving the systems of equations formed by the intersection of the given lines:
- The intersection of 4x + y = 16 and x + y = 10 gives us the point (2,8).
- The intersection of x + y = 10 and the y-axis (x=0) gives us the point (0,10).
- The intersection of 4x + y = 16 and the x-axis (y=0) gives us the point (4,0).
Calculate P for each vertex:
- P at (2,8) = 25(2) + 45(8) = 50 + 360 = 410
- P at (0,10) = 25(0) + 45(10) = 450
- P at (4,0) = 25(4) + 45(0) = 100
The maximum value of P within the feasible region is at point (0,10), resulting in a maximum P value of 450.