mathematics-class-12-unit-12-chapter-05-linear-programming-problems-l-5-6-ye5p7xersg8

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Problem-1 </primary>
<hr>
<main>

-  In order to supplement daily diet, a person wishes to take x and y tablets. The contents (in mg/tab) of iron, calcium and vitamins in X and Y are given as below:

| Tablets  | Iron     | Calcium  | Vitamins |
| -------- | -------- | -------- | -------- |
| X        | 6        | 3        | 2        |
| Y        |2         | 3        | 4        |

- The person needs to supplement atleast $18 \mathrm{mg}$ of iron, $21 \mathrm{mg}$ of Calcium and 16 mg of vitamins. The price of each tablet of X and Y is Rs. 2 & Rs. 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirment at the minimum cost?    
- Make an LPP and solve it graphically.

</div>
<div style='float: right; width:1%;'>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Linear programming lecture-5 $\rightarrow$ <span style = "color:blue"> Problem-1 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Let no. of tablet $X=x$
- no. of tablet $Y=y$
- Let $Z$ be the total cost
- $ Z=2 x+y \text { (Minimize) }$

- Subject to

- $ 6 x+2 y \geqslant 18 \text { ie. } 3 x+y \geqslant 9 \text { (iron constraints) } $

- $ 3 x+3 y \geqslant 21 \text { ie. } x+y \geqslant 7 \text { (calcium constraints) } $

- $ 2 x+4 y \geqslant 16 \text { ie. } x+2y \geqslant 8 \text { (vitamin constraints) } $

- $ x \geqslant 0, y \geqslant 0$

</div>
<div style='float: right; width:1%;'>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Linear programming lecture-5 $\rightarrow$ Problem-1 $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Origin test  for (i) $3 \times 0+0=0 \geqslant 9 \text { (false) }$

- $\therefore$ solution region for  (i) does'nt include origin.

- origin test for (ii) $0+0=0 \geqslant 7 \text { (false) }$

- $\therefore$ Solution region of (ii) does'nt include origin.

- orign test for (iv) $0+2 \times 0=0 \geqslant 8 \text { (false) }$

- origin does not include in solution region.

</div>

</div>
<div style='float: right; width:30%;'>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/a420b4cf-3755-4cca-0934-85489b710d00/public)

</div>
<div style='float: right; width:1%;'>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Corner points are

- A(8,0), B(6,1), C(1,6) \& D(0,9)

- $\therefore$ value of $Z$ at corner points

- $ Z_A=2 \times 8+0=16 $

- $ Z_B=2 \times 6+1=13 $

- $ Z_C=2 \times 1+6=8 \rightarrow$ { Minimum.}

- $ Z_D=2 \times 0+9=9$

</div>
<div style='float: right; width:30%;'>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Problem-2 </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Problem-2 </primary>
<hr>
<main>

-  A dietician has to develop a special diet using two foods $P$ and  $Q$. Each packet of food $P$ contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A.  Each packet of food Q contains 3 unit of calcium ,20 units of iron, 4 units of cholestrol and 3 units of vitamin A.

- The diet requires atleast 240 units 4 calcium, at least 460 units of iron and atmost 300 units of cholestrol.

- How many packets of each food should be used  to minimize the amount of Vit A?   
- What is the minimum amount of Vit A?

</div>
<div style='float: right; width:1%;'>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-2 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Let the no. of packet of food $P and Q$ are $x$ and $y$ respectively.

- Objective function for Vit. A
- $ Z=6 x+3 y \text { (Minimize) }$

- Subject to

- $12 x+3 y \geqslant 240 \text { i.e. } 4 x+y \geqslant 80$ (calcium constraints)

- $4 x+20 y \geqslant 460 \text { i.e. } x+5 y \geqslant 115$ (Iron constraints)

- $6 x+4 y \leqslant 300$ i.e. $3 x+2 y \leqslant 150$ (cholestrol constraints)

- $x \geqslant 0, y \geqslant 0$ (non-negative constraints)

</div>
<div style='float: right; width:1%;'>

</div>
</main>
<hr>
<footer style = "font-size: 18px">Problem-2 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Linear Constraints are

- $
\begin{aligned}
& 4 x+y \geqslant 80 \text { - (i) } \\\\
& x+5 y \geqslant 115 \text { - (ii) } \\\\
& 3 x+2 y \leqslant 150 \text { - (iii) }
\end{aligned}
 $

- Associated eqn for (i), (ii) \& (vii) are

- $ 4 x+y=80 \Rightarrow \frac{x}{20}+\frac{y}{80}=1 $

- $ x+5 y=115 \Rightarrow \frac{x}{115}+\frac{y}{23}=1 $

- $ 3 x+2 y=150 \Rightarrow \frac{x}{50}+\frac{y}{75}=1$

</div>
<div style='float: right; width:1%;'>

</div>
<div style='float: right; width:30%;'>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/35810a3e-f264-4351-2df0-bd356f9d4e00/public)

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- origin test for (i)

- $4 \times 0+0=0 \geqslant 80 \text { (false) }$

- origin does not include in the region.

- origin test tor (ii)

- $0+5 \times 0=0 \geqslant 115 \text { (false) }$

- origin does not include in the  region.

- Origin test for(ii)

- $3 \times 0+2 \times 0=0 \leqslant 150 \text { (true) }$

- $\therefore$ origin iclude in the solution region.

</div>
<div style='float: right; width:1%;'>

</div>
<div style='float: right; width:30%;'>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/b7aa67f3-5533-442f-9a34-13516bdc9200/public)

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Problem-3 </primary>
<hr>
<main>

-  There are two types of fertilizers $A$ and $B$. A consists of $12 \\%$ nitrogen and $5 \\%$. phosphoric acid whereas $B$ consists of $4 \\%$ nitrogen and $5 \\%$ of  phosphoric acid. After testing the soil conditions, farmer finds that he needs atleast 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If $A$ cost Rs 10/kg and B costs Rs 8/kg,  then graphically determine how much of each type of fertilizer should be used so that nutrient requirement are met at a minimum cost?

</div>
<div style='float: right; width:1%;'>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Problem-3 </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Let fertilizer A used = x kg

- Let fertilizer B used = y kg

- Total cost,
- $ Z=10 x+8 y \text { (Minimize) }$

- Subject to

- $ \frac{12 x}{100}+\frac{4 y}{100} \geqslant 12 \text { ie. } 3 x+y \geqslant 300 $        
- $ \frac{5 x}{100}+\frac{5 y}{100} \geqslant 12 \text { ie. } x+y \geqslant 240 $        
- $ x \geqslant 0, y \geqslant 0$

</div>
<div style='float: right; width:1%;'>

</main>
<hr>
<footer style = "font-size: 18px">Problem-3 $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Linear Constraints are

- $ 3 x+y \geqslant 300 ....(i)$             
- $x+y \geqslant 240 ....(ii)$          
- Associated eqn for (i) \& (ii)          
- $3 x+y=300 \Rightarrow \frac{x}{100}+\frac{y}{300}=1$          
- $ x+y=240 \Rightarrow \frac{x}{240}+\frac{y}{240}=1$          
- Origin test  for (i)  $3 \times 0+0=0 \geqslant 300 \text { (False) }$                       
- Origin- does not include          
- origin test for (ii)  $0+0=0 \geqslant 240 \text { (false) }$          
- $\therefore$ orign does not include

</div>
<div style='float: right; width:1%;'>

</div>
<div style='float: right; width:30%;'>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/53b2da2a-e29d-4edf-1ac3-bab4c1f28a00/public)

</div>
<div style='float: right; width:1%;'>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Solution $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Linear Programming Problems L-5 </Success>
### <primary style="font-size:24px"> Solution </primary>
<hr>
<main>

- Half plane $10 x+8 y<1980$ does'nt include any point of open feasible region.

- Min of Z exist.

- $\therefore Z_{\text {min. }}=1980$

- fertilizer $A$ used $=30 \mathrm{~kg}$.      
- fertilizer $B$ used $=210 kg$.

</div>
<div style='float: right; width:30%;'>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ed52edd2-4bc2-4d04-6452-02e56195c000/public)

</div>
<div style='float: right; width:1%;'>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Solution $\rightarrow$ Solution $\rightarrow$ <span style = "color:blue"> Solution </span> $\rightarrow$ Thank you $\rightarrow$  </footer>