Question

A fruit grower can use two types of fertilizer in his garden, brand $P$ and brand $Q$. The amounts (in kg) if nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240$kg of phosphoric acid at least $270$kg of potash and at most $310$ kg of chlorine.
If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

kg per bag

	Brand $P$	Brand $Q$
Nitrogen	$3$	$3.5$
Phosphoric acid	$1$	$2$
Potash	$3$	$1.5$
Chlorine	$1.5$	$2$

Answer 1

Let the fruit grower use $x$ bags of brand $P$ and $y$ bags of brand $Q$.

The problem can be formulated as follows:

Minimize $z=3x+3.5y........\left(1\right)$

subject tot he constraints

$x+2y\ge 240.......\left(2\right)$

$x+0.5\ge 90......\left(3\right)$

$1.5x+2y\le 310......\left(4\right)$

$x,y\ge 0........\left(5\right)$

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

The corner points are $A(240,0),B(140,50)$ and $C(20,140)$

The values of $z$ at these corner points are as follows.

Corner point $z=3x+3.5y$

$A(140,50)$ $595$

$B\left(20.140\right)$ $550$

$C(40,100)$ $470$ $\to$ Minimum

The maximum value of $z$ is $470$ at $(40,100)$

Thus, $40$ bags of brand $P$ and $100$ bags of brand $Q$ should be added to the garden to minimize the amount of nitrogen.

The minimum amount of nitrogen added to the garden is $470$ kg.