Question

A store sells two types of toys, $A$ and $B$. The store owner pays 8$and$ $14$ for each one unit of toy $A$ and $B$ respectively. One unit of toys $A$ yields a profit of 2$whileaunitoftoys$B$yieldsaprofitof$ $2$. The store owner estimates that no more than $2000$ toys will be sold every month and he does not plan to invest more than 20,000$ in inventory of these toys. How many units each type of toys should be stocked in order to maximize his monthly total profit?

Answer 1

Let $x$ be the total number of toys $A$ and $y$ the number of toys $B$; $x$ and $y$ cannot be negative, hence
$x\ge 0$ and $y\ge 0$
The store owner estimates that no more than $2000$ toys will be old every month.
$x+y\le 2000$
One unit of toys $A$ yields a profit of 2$whileaunitoftoys$B$yieldsaprofitof$ $3$, hence the total profit $P$ is given by
$P=2x+3y$
The store owner pays 8$and$ $14$ for each one unit of toy $A$ and $B$ respectively and he does not plant to invest more than 20,000$ in inventory of these toys
$8x+14y\le 20,00$
What do we have to solve?
Find $x$ and $y$ so that $P=2x+3y$ is maximum under the conditions
$\{\begin{array}{c}x\ge 0\\ x\ge 0\\ x+y\le 2000\\ 8x+14y\le 20,000\end{array}$
The solution set of the system of inequalities given above and the vertices of the region obtained are shown below:
Vertices of the solution set
$A$ at $(0,0)$
$B$ at $\left(0.1429\right)$
$C$ at $(1333,667)$
$D$ at $(2000,0)$
Calculate the total profit $P$ at each vertex
$P\left(A\right)=2\left(0\right)+3\left(0\right)=0$
$P\left(B\right)=2\left(0\right)+3\left(1429\right)=4287$
$P\left(C\right)=2\left(1333\right)+3\left(667\right)=4667$
$P\left(D\right)=2\left(2000\right)+3\left(0\right)=4000$
The maximum profit is at vertex $C$ with $x=1333$ and $y=667$
Hence the store owner has to have $1333$ toys of type $A$ and $667$ toys of type $B$ in order to maximize his profit