Hello, Bluewaterjl415!
The cost to a store for two models of computers are $900 and $1075.
The $900 model yeilds a profit of $100 and the $1075 model yeilds a profit of $125.
Market tests and availible resources have indicated the following constraints:
- the merchant estimates that the total monthly demand will not exceed 275 units.
- the merchant does not want to invest more than $260,625 in computer inventory.
What is the optimal inventory level for each model?
What is the optimal profit?
Let \(\displaystyle x\) = number of $900 models stocked. \(\displaystyle \,x\,\geq\,0\)
\(\displaystyle \;\;\)They will cost \(\displaystyle 900x\) dollars.
Let \(\displaystyle y\) = number of $1075 models stocked. \(\displaystyle \,y\,\geq\,0\)
\(\displaystyle \;\;\)They will cost \(\displaystyle 1075y\) dollars.
The total cost must not exceed $260,625: \(\displaystyle \:900x\,+\,1075y\:\leq\:260,625\;\)
[1]
The monthly demand is at most 275 units.
The total number of computers must not exceed 275: \(\displaystyle \;x\,+\,y\:\leq\:275\;\)
[2]
Graph and shade the region determined by the inequalties.
We get a quadrilateral in quadrant 1.
It has vertices: \(\displaystyle (0,0),\;(275,0),\;\left(0,\frac{10425}{43}\right)\)
\(\displaystyle \;\;\)and a fourth vertex, the intersection of the two lines.
Solving: \(\displaystyle \,\begin{bmatrix}900x\,+\,1075y\:=\:260625\\x\,+\,y\:=\:0\end{bmatrix}\) . . . we get: \(\displaystyle \,x\,=\,200,\;y\,=\,75\)
Test each vertex in the profit function: \(\displaystyle \,P\:=\:100x\,+\,125y\)
\(\displaystyle \;\;\)and see which one produces maximum profit.
