Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs. 60 per kg and food Q costs Rs. 80 per kg. Food P contains 3 units per kg of vitamin A and 5 units per kg of vitamin B while food Q contains 4 units per kg of vitamin A and 2 units per kg of vitamin B. Determine the minimum cost of the mixture.

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Solution

Let Reshma mixes x kg of food P and y kg of food Q. Construct the following table:

$\begin{matrix} F o o d & Q u a n t i t y & V i t a m i n A & V i t a m i n B & C o s t (R s . p e r k g) P & x k g & 3 x & 5 x & 60 x Q & y k g & 4 y & 2 y & 80 y T o t a l & 3 x + 4 y & 5 x + 2 y & 60 x + 80 y R e q u i r e m e n t & A t l e a s t 8 & A t l e a s t 11 \end{matrix}$

The mixture must contain atleast 8 units of vitamin A and 11 units of vitamin B, Total cost Z of purchasing food is Z = 60x + 80y

The mathematical formulation of the given problem is

Minimize Z = 60x + 80y ............(i)

Subject to the constrains 3x + 4y $\geq$ 8 .........(ii)

5x + 2y $\geq$ 11 .........(iii)

x $\geq$ 0, y $\geq$ 0 .......(iv)

Firstly, draw the graph of the line 3x + 4y = 8

$\begin{matrix} x & 0 & \frac{8}{3} y & 2 & 0 \end{matrix}$

Putting (0, 0) in the inequality 3x + 4y $\geq$ 8, we have

$3 \times 0 + 4 \times 0 \leq 8$

$\Rightarrow 0 \geq 8$ (which is false)

So, the half plane is away from the origin.

Since, x, y $\geq$ 0

So, the feasible region lies in the first quadrant.

Secondly, draw the graph of the line 5x + 2y = 11

$\begin{matrix} x & 0 & \frac{11}{5} y & \frac{11}{2} & 0 \end{matrix}$

Putting (0, 0) in the inequality 5x + 2y $\geq$ 11,

we have $5 \times 0 + 2 \times 0 \geq 11$

$\Rightarrow 0 \geq 11$ (which is false)

So, the half plane is away from the origin.

It can be seen that the feasible region is unbounded.

On solving equations $3 x + 4 y = 8$ and 5x + 2y = 11, we get $B (2, \frac{1}{2})$

The corner points of the feasible region are $A (\frac{8}{3}, 0) B (2, \frac{1}{2}) a n d C (0, \frac{11}{2})$ .

The values of Z at these points are follows :

$\begin{matrix} C o r n e r p o i n t & Z = 60 x + 80 y A (\frac{8}{3}, 0) & 160 \to M i n i m u m B (2, \frac{1}{2}) & 160 \to M i n i m u m C (0, \frac{11}{2}) & 440 \end{matrix}$

As the feasible region is unbounded, therefore 160 may or may not be the minimum value of Z. For this, we graph the inequality 60 x + 80 y < 160 or 3x + 4y < 8 and check whether the resulting half plane has feasible region has no common point with 3x + 4y < 8 therefore, the minimum cost of the mixture will be Rs. 160 at the line segment joining the points $A (\frac{8}{3}, 0) a n d B (2, \frac{1}{2})$ .

Note. Please be careful, while making the table and plotting the graph of inequalities.

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Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units /kg of vitamin A and 5 units/kg of vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B.Let the mixture contain x kg of food P and y kg of food Q. Which of the following are the constraints of this linear programming problem?

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