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 .

Question

a. Identify the vertices of the feasible region.
b. What production levels yield the maximum profit, and what is the maximum profit?

Answer 1

To find the vertices of the feasible region, we first need to solve the inequality x + 2y ≤ 1400 given the additional constraint that y must exceed x by at least 100 units.

1. Start by rewriting the inequality with the constraint y - x ≥ 100:
x + 2(x + 100) ≤ 1400
x + 2x + 200 ≤ 1400
3x ≤ 1200
x ≤ 400

2. Given the constraint y - x ≥ 100:
y ≥ x + 100
y ≥ 400 + 100
y ≥ 500

So the feasible region is bounded by x ≤ 400, y ≥ 500, and x + 2y ≤ 1400. The vertices of the feasible region can be found by solving the system of equations created by the boundaries.

1. (x = 400, y = 500)
2. (x = 400, y = 500)
3. (x = 200, y = 600)
4. (x = 0, y = 700)
5. (x = 0, y = 500)

To find the production levels that yield the maximum profit, we evaluate the profit function P = 14x + 22y - 900 at each of the vertices of the feasible region:
1. P(400, 500) = 14(400) + 22(500) - 900 = 8800
2. P(400, 600) = 14(400) + 22(600) - 900 = 9500
3. P(200, 600) = 14(200) + 22(600) - 900 = 8300
4. P(0, 700) = 14(0) + 22(700) - 900 = 14400
5. P(0, 500) = 14(0) + 22(500) - 900 = 10100

Therefore, the maximum profit is $14400 when x = 0 and y = 700.