a factory can produce two products, x and y with a profit approximated by P=14x+22y-900. the production of y can exceed x by no more than 100 units moreover production levels are limited by the formula x+2y,+1,400. what production levels yield maximum profit?
What does this constraint mean?
limited by the formula x+2y,+1,400.
I have to leave, hope Steve sees this.
this question makes me so confused...
To find the production levels that yield the maximum profit, we need to maximize the profit function P = 14x + 22y - 900, while adhering to the given constraints:
1. The production of y cannot exceed x by more than 100 units: y <= x + 100
2. The production levels are limited by the formula x + 2y <= 1,400
To solve this problem, we can use the method of linear programming. Linear programming involves finding the maximum or minimum value of a linear objective function, subject to a set of linear constraints.
Step 1: Rewrite the constraints in standard form:
y - x <= 100 (multiply both sides by -1)
x + 2y <= 1,400
Step 2: Plot the feasible region defined by the constraints. The feasible region is the area where all constraints are satisfied.
- Plot the line y - x = 100 (draw a dashed line because it is an inequality, and shade the region below the line).
- Plot the line x + 2y = 1,400 (draw a dashed line because it is an inequality, and shade the region below the line).
- The feasible region is the area where the two shaded regions overlap.
Step 3: Calculate the coordinates of the vertices of the feasible region. The vertices are the points where the lines that define the feasible region intersect.
Step 4: Substitute the coordinates of each vertex into the profit function P = 14x + 22y - 900 and calculate the profit for each combination of production levels.
Step 5: Determine the combination of production levels that yields the maximum profit.
By following these steps, you can find the production levels that will yield the maximum profit.
Constraints:
A. x≥0
B. y≥0
C. y-x≤100
D. x+2y≤1400 (please check this constraint)
Draw the graph of the constraint equation to have a visual understanding.
Solve for all possible corners of the feasible polygon. Note that all inequalities are inclusive, i.e. all lines are part of the feasible region.
A/B: (0,0)
A/C: (0,100)
A/D: - (y-x>100)
B/C: - (x<0)
B/D: (1400,0)
C/D: (650,750)
Since the feasible polygon is necessarily convex, we only have to check profit at the intersections.
P(x,y)=14x+22y-900
P(0,100)=1300
P(1400,0)=18700
P(650,750)=24700
So the maximimum profit can be attained by X=650, Y=750.