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?

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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 = 3x + 3.5y … (1)

subject to the constraints,

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.

Corner point

z = 3x + 3.5y

A (140, 50)

595

→ Maximum

B (20, 140)

550

C (40, 100)

470

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.

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Q. If the grower wants to maximise 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?

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of nitrogen, phosphoric acid, potash and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs atleast 240 kg of phosphoric acid, atleast 270 kg of potash and atmost 310 kg of chlorine. 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?

$\begin{matrix} B r a n d P & B r a n d Q N i t r o g e n & 3 & 3.5 P h o s p h o r i c a c i d & 1 & 2 P o t a s h & 3 & 1.5 C h l o r i n e & 1.5 & 2 \end{matrix}$

Q. 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 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?

kg per bag

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

Q. A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of 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 Phosphoric acid Potash Chlorine 3 1 3 1.5 3.5 2 1.5 2

A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts (in kg) of 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 atmost 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?

$\begin{matrix} k g p e r b a g \begin{matrix} F e r t i l i z e r & B r a n d P & B r a n d Q N i t r o g e n & 3 & 3.5 P h o s p h o r i c a c i d & 1 & 2 P o t a s h & 3 & 1.5 C h l o r i n e & 1.5 & 2 \end{matrix} \end{matrix}$

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Corner point	z = 3x + 3.5y
A (140, 50)	595	→ Maximum
B (20, 140)	550
C (40, 100)	470