Question

A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per heclare are estimated as Rs 10,500 and Rs 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit?
From an LPP from the above and solve it graphically. Do you agree with the massage that the protection of wildlife is utmost necessary to preserve the balance in environment?

Answer 1

Let $x$ hectares fro crop $A$ and $y$ hectares for crop $B$ be allocated.

$\text{A}$	$\text{B}$
$x$	$y$	$50$ hectares
$10500$	$9000$	Profit
$20L/H$	$10L/H$	$800$ litres atmost

We need to maximize profit given by

$Z=10500x+9000y$ subject to

$x+y\le 50$

$20x+10y\le 800$

$x,y\ge 0$

After plotting graph, the corner points are

Corner Point	Value of $Z$
$O(0,0)$ $A(40,0)$ $B(30,20)$ $C(0,50)$	$Z=0$ $Z=420000$ $Z=315000+180000=495000$ $Z=450000$	Maximum

Therefore, $30$ hectares for crop $A$ and $20$ hectares for crop $B$.