Question 1147365: Josh runs a bakery that sells two kinds of desserts. Josh knows the bakery must make at least 16 and at most 60 boxes of the Fluffy Deliciousness. The bakery must also make between 11 and 65 boxes of the White Chocolate Blizzards. The boxes of Fluffy Deliciousness take 13 ounces of sugar, while boxes of White Chocolate Blizzards require 12 ounces of sugar. The bakery only has 1404 ounces of sugar available. If boxes of Fluffy Deliciousness generate $1.08 in revenue, and boxes of White Chocolate Blizzards generate $1.53, how many boxes of the desserts should Josh have the bakery make to get the most revenue?
find
Fluffy Deliciousness:
White Chocolate Blizzards:
Best revenue:
Answer by greenestamps(13216)   (Show Source):
You can put this solution on YOUR website!
Let x be the number of Fluffy Deliciousness
Let y be the number of White Chocolate Blizzards
The defined constraints are
(1) 16 <= x <= 60 (green lines)
(2) 11 <= y <= 65 (blue lines)
(3) 13x+12y <= 1404 (red line)
A picture....
The formal mathematics says the maximum revenue is at one of the corners of the feasibility region, A, B, C, D, or E. But common sense tells us that the maximum revenue will be at either C or D.
Solve for the coordinates of C by solving 13x+12y=1404 with y=65; solve for the coordinates of D by solving 13x+12y=1404 with x=60.
Then evaluate the objective function, 1.08x+1.53y, at those two points.
Note there is a refinement of the general process, seldom if ever mentioned in references, that can make it easier and faster to get to the final answer.
Specifically, it is NOT necessary to evaluate the objective function at every corner of the feasibility region. The corner of the feasibility region that will generate the maximum revenue can be determined by comparing the slopes of the constraint lines to the slope of the objective function.
In this problem there is only one slanted constraint line; its slope is -13/12;
The slope of the objective function is -1.08/1.53;
So the slope of the objective function is less negative than the slope of the constraint line.
The objective function will be maximized when its graph just touches a corner of the feasibility region. Since the slope of the objective function is less negative than the slope of the constraint line, the maximum revenue will be obtained at C.
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