A factory can produce two products, x and y, with a profit approximated by P=14x+22y−900 . The production of y must exceed the production of x by at least 100 units. Moreover, production levels are limited by the formula x+2y≤1400 .Identify the vertices of the feasible region.What production levels yield the maximum profit, and what is the maximum profit?
To find the vertices of the feasible region, we need to solve the system of inequalities:
1) y > x + 100 (production of y must exceed the production of x by at least 100 units)
2) x + 2y ≤ 1400 (production levels are limited by this formula)
First, let's solve inequality 1 for y and substitute it into inequality 2:
y > x + 100
x + 2(x + 100) ≤ 1400
x + 2x + 200 ≤ 1400
3x ≤ 1200
x ≤ 400
So we have x ≤ 400.
Now let's solve inequality 2 for y:
x + 2y ≤ 1400
2y ≤ 1400 - x
y ≤ (1400 - x)/2
Substituting x = 400, we get:
y ≤ (1400 - 400)/2
y ≤ 500
So we have y ≤ 500.
The vertices of the feasible region are found at the intersections of the lines x = 400 and y = 500 with the line x + 2y = 1400.
The intersection points are:
1) (400, 300)
2) (400, 500)
3) (600, 400)
Next, let's find the production levels that yield the maximum profit and the maximum profit itself.
To do this, we need to evaluate the profit function at each vertex:
P(400, 300) = 14(400) + 22(300) - 900 = 5600 + 6600 - 900 = 11300
P(400, 500) = 14(400) + 22(500) - 900 = 5600 + 11000 - 900 = 16200
P(600, 400) = 14(600) + 22(400) - 900 = 8400 + 8800 - 900 = 16300
Therefore, the maximum profit is $16,300, and it is achieved at the production levels of x = 600 and y = 400.