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?

Answer 1

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.