Hello, jfsfroggy!
Are you completely new to Linear Programming?
Two kinds of crated cargo, A and B, are to be shipped by truck.
The weight and volume of each type are given in the following table:
Code:
A B
Volume | 50 ft³ | 10 ft³ |
Weight | 200 lbs | 360 lbs |
The shipping company charges $75 per crate for cargo A and $100 per crate for cargo B.
The truck has a maximum load limit of 7200 pounds and 1000 cubic feet.
How many of each type of cargo should be shipped to maximize profit?
Let \(\displaystyle x\) = number of crates of \(\displaystyle A\).
They will require: \(\displaystyle 50x\) ft<sup>3</sup> of space and will weigh \(\displaystyle 200x\) pounds.
Let \(\displaystyle y\) = number of crates of \(\displaystyle B\).
They will require: \(\displaystyle 10y\) ft<sup>3</sup> of space and will weigh \(\displaystyle 360y\) pounds.
The total volume has a maximum of 1000 ft<sup>3</sup>: \(\displaystyle \,50x\,+\,10y\:\leq\:1000\;\)
[1]
The total weight has a maximum of 7200 pounds: \(\displaystyle \,200x\,+\,360y\:\leq\:7200\;\)
[2]
\(\displaystyle \;\;\)Divide [1] by 10: \(\displaystyle \:5x\,+\,y\:\leq\:100\)
\(\displaystyle \;\;\)Divide [2] by 40: \(\displaystyle \:5x\,+\,9y\:\leq\:180\)
Graph these inequalities in the first quadrant.
The first inequality is the region below the line with intercepts (20,0) and (0,100).
The second is the region below the line with intercepts (36,0) and (0,20).
\(\displaystyle \;\;\)Solve the two
equations to find their intersection: (18,10).
We have a quadrilateral with vertices: (0,0), (20,0), (18,10), (0,20).
Test these in the Profit Function: \(\displaystyle \
\;=\;75x\,+\,100y\)
\(\displaystyle \;\;\)and determine which one produces maximum profit.
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