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?

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Solution

Let's assume that bags of brand P needed be $X$ and brand Q be $Y$

Since bag of brand P contains 1 kg and bag of brand Q contains 2 kg of Phosphoric acid. Also, garden needs at least 240 kg of phosphoric acid.
$∴ X + 2 Y \geq 240 . . . (1)$

Since bag of brand P contains 3 kg and bag of brand Q contains 1.5 kg of potash. Also, garden needs at least 270 kg of potash.
$∴ 3 X + 1.5 Y \geq 270$

$\Rightarrow 2 X + Y \geq 180 . . . (2)$

Since bag of brand P contains 1.5 kg and bag of brand Q contains 2 kg of chlorine. Also, garden needs at most 310 kg of chlorine.
$∴ 1.5 X + 2 Y \leq 310 . . . (3)$

Since count of bags can't be negative.
$∴ X \geq 0, Y \geq 0 . . . (4)$

Now, Bag of brand P contains 3 kg of Nitrogen and Bag of brand Q contains 3.5 kg of Nitrogen.

So, total Nitrogen contents $Z = 3 X + 3.5 Y$

We have to maximise the amount of Nitrogen added to the garden.

After plotting all the constraints given by equation (1), (2), (3) and (4) we get the feasible region as shown in the image.

Corner points Value of $Z = 3 X + 3.5 Y$
A (20, 140) 550
B (40, 100) 470
C (140, 50) 595 (maximum)
Hence, to maximise the amount of nitrogen added to the garden, grower should use $140$ bags of brand P and $50$ bags of brand Q. In this case maximum amount of nitrogen added to the garden will be $595 K g$

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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?

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) 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

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	$1.5$	$2$

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 points	Value of $Z = 3 X + 3.5 Y$
A (20, 140)	550
B (40, 100)	470
C (140, 50)	595 (maximum)