Question

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?

Answer 1

Answers:

a) vertices: (400, 500); (0, 100); (1400,0)

b) production levels that yield the maximum profit and maximum profit:

x = 1400, y = 0, P = 18,700.

Explanation

Part A.

1) State the restrictions:

i) The production of y must exceed the production of x by at least 100 units:

⇒ y ≥ x + 100

ii) Production levels are limited by the formula x+2y ≤ 1400.

⇒ y ≤ 700 - x/2

iii) x and y cannot be negative:

⇒ x ≥ 0 and y ≥ 0

2) The feasible region is the area inside the lines defined by:

y = x + 100, y = 700 - x/2, y = 0, and x = 0

3) The vertices are found by solving 3 separated systems of equations:

i) intersection of lines y = x + 100 and y = 700 - x/2

⇒ x + 100 = 700 - x/2

⇒ 3x/2 = 600 ⇒ x= 1200 / 3 = 400

⇒ y = 400 + 100 = 500

⇒ vertex = (400, 500)

ii) intersection of lines y = x + 100 and x = 0

⇒ x = 0 and y = 100

⇒ vertex = (0, 100)

iii) intersection of lines y = 700 - x/2 and y = 0

⇒ y = 0 ⇒ x/2 = 700 ⇒ x = 1400

⇒ vertex = (1400, 0)

Therefore the vertices of the feasible region are:

(400, 500); (0, 100); (1400,0)

Part B.

The production levels that yield the maximum profit, and the maximum profit are found by replacing the vertices in the profit equation. The vertex that yields the maximum profit is the solution

1) vertex (400, 500)

P = 14x + 22y - 900 = 14(400) + 22(500) - 900 = 15,700

2) vertex (0, 100)

P = 14x + 22y - 900 = 14(0) + 22(100) - 900 = 1,300 (clearly less profit)

3) vertex (1400, 0)

P = 14x + 22y - 900 = 14(1400) + 22(0) - 900 = 18,700, which is the highest profit.

Answer 2

We know that 
$x+100 \leq y x+2y \leq 1400$

To solve a. you just need to equalize those two inequalities:
x+2y=1400
y=x+100

x+2(x+100)=1400 => 3x=1200 => x=400
y=500

b. To find the maximum profit put the values you got instead of x and y.
p=14*400+22*500-900 = 5600 + 11000-900 = 15700

I hope that this is the answer that you were looking for and it has helped you.