# Refer to question 8.

Question:

Refer to question 8. If the grower wants to maximize the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?

Solution:

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

The problem can be formulated as follows.

Maximize $z=3 x+3.5 y$                 (1)

subject to the constraints,

$x+2 y \geq 240$                            (2)

$x+0.5 y \geq 90$                           (3)

$1.5 x+2 y \leq 310$                       (4)

$x, y \geq 0$                                  (5)

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

The corner points are A (140, 50), B (20, 140), and C (40, 100).

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

The maximum value of z is 595 at (140, 50).

Thus, 140 bags of brand P and 50 bags of brand Q should be used to maximize the amount of nitrogen.

The maximum amount of nitrogen added to the garden is 595 kg.