Question

A factory makes tennis rackets and cricket bats. A tennis racket takes $1.5$ hours of machine time and $3$ hours of craftsman's time in its making while a cricket bat takes $3$ hour of machine time and $1$ hour of craftsman's time. In a day, the factory has the availability of not more than $42$ hours of machine time and $24$ hours of craftsman's time.

If the profit on a racket and on a bat is Rs.$20$ and Rs.$10$ respectively, find the maximum profit of the factory when it works at full capacity

Answer 1

The given information can be compiled in a table as follows.

	Tennis Racket	Cricket Bat	Availability
Machine Time (h)	$1.5$	$3$	$4.2$
Craftsman's Time (h)	$3$	$1$	$24$

$\therefore$ $1.5x+3y\le 42$
$3x+y\le 24$
$x,y\ge 0$
The profit on a racket is Rs.$20$ and on a bat is Rs.$10$
$\therefore$ $Z=20x+10y$
The mathematical formulation of the given problem is
Maximise $Z=20x+10y.......\left(1\right)$
subject to the constraints
$1.5x+3y\le \mathrm{42..........}\left(2\right)$
$3x+y\le \mathrm{24............}\left(3\right)$
$x,y\ge \mathrm{0..........}\left(4\right)$

The feasible region determined by the system of constraints is as shown.

The corner points are $A(8,0),B(4,12),C(0,14)$ and $0(0,0).$

The values of Z at these corner points are tabulated as shown in above figure

REF. IMAGE 2

The maximum value of Z is 200, which occurs at $x=4$ and $y=12.$

Thus, the factory must produce $4$ tennis rackets and $12$ cricket bats to earn the maximum profit. The maximum profit earned by the factory will be Rs $200.$