SOLUTION: 2 A small manufacturer produces two kinds of good, A and B, for which demand exceeds capacity. The production costs for A and B are $6 and $3, respectively, each, and the correspon

Question 1071240: 2 A small manufacturer produces two kinds of good, A and B, for which demand exceeds capacity. The production costs for A and B are $6 and $3, respectively, each, and the corresponding selling prices are $7 and $4. In addition, the transport costs are 20 cents and 30 cents for each good of type A and B, respectively. The conditions of a bank loan limit the manufacturer to maximum weekly production costs of $2700 and maximum weekly transport costs of $120.
How should the manufacturer arrange production to maximize profit?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
x = number of units of product A.
y = number of units of product B.

your constraint inequalitiess are:

x >= 0
y >= 0
6x + 3y <= 2700
.2x + .3y <= 120

selling price is equal to 7x + 4y

profit is equal to 7x - 6x - .2x + 4y - 3y - .3y

combine like terms and you get:

profit is equal to .8x + .7y

that's your objective function.

using the desmos.com calculator, you would graph the opposite inequalities.

to be more specific, you would graph the following inequalities.

x <= 0
y <= 0
6x + 3y >= 2700
.2x + .3y >= 120

the area of the graph that is NOT shaded is your region of feasibility.

your graph will look like this.

$$$

you then find the corner points of this region and evaluate your objective function at those corner points.

you will get.

profit at (0,400) = 280
profit at (375,150) = 405
profit at (450,0) = 360

your maximum profit is at (375,150).

you have to meet your constraints at those corner points.

production costs at (375,150) = 6x + 4y = 2700 which meets the constraint that they be less than or equal to 2700.

transport costs at (375,150) = .2x + .3y = 120 which meets the constraint that they be less than or equal to 120.

x >= 0 and y >= 0 are also meet the constraint that they be greater than or equal to 0 at the point (375,150).

all constraints are met.

your maximum profit is when you sell 375 units of product A and 150 units of product B.

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