## Book: Mathematics Part-II

### Chapter: 12. Linear Programming

#### Subject: Maths - Class 12th

##### Q. No. 1 of Exercise 12.2

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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. Determine the minimum cost of the mixture.

Let the mixture consists of x kg of food P and y kg of food Q. x ≥ 0 and y ≥ 0

According to given situation in the question, the information can be tabulated as follows:

The mixture must contain at least 8 units of vitamin A and 11 units of vitamin B.

The constraints are:

3x + 4y ≥ 8 and

5x + 2y ≥ 11.

Where x ≥ 0 and y ≥ 0

Total cost of ‘Z’ of purchasing food is Z= 6ox+ 80y

The mathematical formulation of the given problem is: minimize Z= 6ox + 80y…….1

Subject to constraints

3x + 4y ≥ 8 and

5x + 2y ≥ 11.

Where x , y ≥ 0

The graphical representation shows the feasible region is unbounded.

The corner points of the feasible region are

The values of Z at these corner points are

As the feasible region is unbounded

160 may or may not be the minimum value of Z

But it can be seen that the feasible region has no common oint with 3x+4y <8

So the minimum cost of the mixture will be Rs160 at the line segment joining the points

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