SOLUTION: A toy manufacture makes 2 kinds of toy wagons, a standard wagon and a deluxe. The standard wagon requires 10 labor hours for assembly and 10 hours for finishing. The deluxe require

Question 1125190: A toy manufacture makes 2 kinds of toy wagons, a standard wagon and a deluxe. The standard wagon requires 10 labor hours for assembly and 10 hours for finishing. The deluxe require 10 hours for assembly and 20 hours for finishing. The factory can provide up to 300 hours per week for assembly and up to 400 hours for finishing. Each standard wagon yields a profit of $10 and each deluxe wagon yields 25$. How many wagons of each type should be made per week to maximize the total profit?
Answer by greenestamps(13216) (Show Source): You can put this solution on YOUR website!

Let x be the number of standard wagons and y be the number of deluxe wagons.

Each standard and each deluxe wagon requires 10 hours of assembly; the maximum number of hours available for assembly is 300:

Each standard wagon requires 10 hours of finishing; each deluxe wagon requires 20 hours of finishing. The maximum number of hours available for finishing is 400:

Some algebra shows that the corners of the feasibility region resulting from those two constraints are (0,0), (0,20), (20,10), and (30,0).

The profit function ("objective function") is 10x+25y = P. Evaluating that objective function at the corners of the feasibility region gives

(0,0): profit = 0+0 = 0
(0,20): profit = 0+500 = 500
(20,10): profit = 200+250 = 450
(0,30): profit = 0+750 = 750

The maximum profit is when 30 deluxe wagons are built and no standard wagons.

Most resources will tell you that you need to find all the corners of the feasibility region and evaluate the objective function at every corner to find the maximum value of the objective function.

That is not true. You can tell which corner of the feasibility region will give the maximum value of the objective function by comparing the slopes of the constraint boundary lines to the slope of the objective function.

In slope-intercept form, the constraint boundary lines for this problem are

(slope -1) and
(slope -1/2)

The equation of the objective function is

(slope -5/2)

The maximum value of the objective function will be at the corner of the feasibility region where a line with slope -5/2 will just touch the feasibility region and not pass through it.

Simple analysis shows that the corner of the feasibility region where that happens is (30,0).

By using this reasoning to solve the problem, we don't even have to find where the two constraint lines intersect, because we know that is not where the maximum value of the objective function will be obtained.
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