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