Question

A dealer in rual area wishes to purchase a number of sewing machines. He has only $Rs5,760$ to invest and has space for at most $20$ items for storage. An electronic sewing machine costs him $Rs360$ and a manually operated sewing machine $Rs240$. He can sell an electronic sewing machine at a profit of $Rs22$ and a manually operated sewing machine at a profit of $Rs18$. Assuming that he can sell all the items that he can buy how should he invest his money in order to maximize his profit? Formulate this problem mathematically and then solve it.

Answer 1

Let $x$ and $y$ be electronic and manually operated sewing machines purchased respectively.
Therefore, LPP is maximize $P=22x+18y$
Subjected to

$\begin{array}{c}360x+240y\le 5760\\ or,3x+2y\le 48\\ x+y\le 20\\ x\ge 0,y\ge 0\end{array}\}$

For correct graph vertices of feasible region are
$A(0,20),B(8,12),C(16,0)andO(0,0)$
$P\left(A\right)=360,P\left(B\right)=392,P\left(C\right)=352$
$3x+2y=48--\left(1\right)$
$x+y=20---\left(2\right)$

Solving equation $\left(1\right)$ and $\left(2\right),$ we get
$x=8,y=12$

Therefore for maximum $P,$ electronic machines $=8.$

Manual machines $=12.$