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 hectare 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 and other wildlife is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is utmost necessary to preserve the balance in environment?

Answer 1

Let the land allocated to crop $X$ be $x$ hector and land allocated to crop $Y$ be $y$ hector.

Given: Herbicides used for crop$X=20$ litres per hectare

Herbicides used for crop$Y=10$ litres per hectare

Maximum quantity of herbicide=$800$ litres

$\therefore 20x+10y\le 800$

$2x+y<800$

The total land available =$50$ hectare

$\therefore x+y\le 50$

As we count to maximise the profit

Hence, the function used here is Maximise z

Profit from Crop $x=$ Rs.$10500$per hectare

Profit from Crop $y=$ Rs.$9000$ per hectare

Maximize $z:10500x+9000y$

Combining all constraints $2xy\le 80$

$x+y\le 50$

$x,y\ge 0$

i) $x+y\le 50$

$x$	0	50
$y$	50	0

ii) $2x+y\le 80$

$x$	0	40
$y$	80	0

Corner Points	Value of z
(0,50)	450000
(30,20)	495000 $\to$ maximum
(40,0)	420000
(0,0)	0

Hence the profit will be maximum of

Crop $x=30$ hectare

Crop $y=20$ hectare

Maximum profit $=$ Rs.$4,95,000$