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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Solution

Let the fruit grower mixes x bags of brand P and y bags of brand Q. Construct the following table:

$\begin{matrix} B r a n d o f & N u m b e r o f & A m o u n t o f & A m o u n t o f & A m o u n t o f & A m o u n t o f f e r t i l i z e r & b a g s & n i t r o g e n & p h o s p h o r i c a c i d & p o t a s h & c h l o r i n e P & x & 3 x & x & 3 x & 1.5 x Q & y & 3.5 y & 2 x & 1.5 y & 2 y T o t a l & x + y & 3 x + 3.5 y & x + 2 y & 3 x + 1.5 y & 1.5 x + 2 y \end{matrix} O u r p r o b l e m i s t o m i n i m i z e Z = 3 x + 3.5 y . . (i) S u b j e c t t o c o n s t r a i n t s a r e x + 2 y \geq 240 . . (i i) 3 x + 1.5 y \geq 270 . . (i i i) 1.5 x + 2 y \leq 310 . . (i v) x \geq 0, y \geq 0 . . (v) F i r s t l y, d r a w t h e g r a p h o f t h e l i n e x + 2 y = 240 \begin{matrix} x & 0 & 240 y & 120 & 0 \end{matrix} P u t t i n g (0, 0) i n t h e i n e q u a l i t y x + 2 y \geq 240, w e h a v e 0 + 2 \times 0 \geq 240 \Rightarrow 0 \geq 240 (w h i c h i s f a l s e) S o, t h e h a l f p l a n e i s a w a y f r o m t h e o r i g i n . S e c o n d l y, d r a w t h e g r a p h o f t h e l i n e 3 x + 1.5 y = 270 \begin{matrix} x & 0 & 90 y & 180 & 0 \end{matrix} P u t t i n g (0, 0) i n t h e i n e q u a l i t y 3 x + 1.5 y \geq 270, w e h a v e 3 \times 0 + 1.5 \times 0 \geq 270 \Rightarrow 0 \geq 270 (w h i c h i s f a l s e) S o, t h e h a l f p l a n e i s a w a y f r o m t h e o r i g i n . T h i r d l y, d r a w t h e g r a p h o f t h e l i n e 1.5 x + 2 y = 310 \begin{matrix} x & 0 & 620 / 3 y & 155 & 0 \end{matrix} P u t t i n g (0, 0) i n t h e i n e q u a l i t y 1.5 x + 2 y \leq 310, w e h a v e 1.5 \times 0 + 2 \times 0 \leq 310 \Rightarrow 0 \leq 370 (w h i c h i s t r u e) S o, t h e h a l f p l a n e i s t o w a r d s t h e o r i g i n .$

Since, $x, y \geq 0$

So, the feasible region lies in the first quadrant.

Let Z be the total cost.

On solving equations 1.5x+2y=310 and x+ 2y = 240, we get

A(140, 50)

Similarly, on solving equations 3x + 1.5y = 270 and 1.5x + 2y = 310,

we get B(20, 140)

$∴$ Feasible region is ABCA.

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

$\begin{matrix} C o r n e r p o i n t & Z = 3 x + 3.5 y A (140, 50) & 595 B (20, 140) & 550 C (40, 100) & 470 \to M i n i m u m \end{matrix}$

The minimum value of Z is 470 at C(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.

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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 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?

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