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A Dietician Wishes to Mix Two Kinds ·Of Food X· and Y in Such a Way that The Mixture Contains at Least 10 Units of Vitamin A, 12 Units of Vitamin B Arid 8 Units of Vitamin C. the Vitamin Contents of One Kg Food is Given Below - Mathematics

A dietician wishes to mix two kinds ·of food X· and Y in such a way that the  mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:

Food Vitamin A Vitamin.B Vitamin C
X 1 unit 2 unit 3 unit
Y 2 unit 2 unit 1 unit

Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.

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Solution

Let one type of food be x and another type be y

Food Vitamin A Vitamin B Vitamin C
X 1 units 2 units 3 units
Y 2 units 2 units 1 units

∴ According to the given condition

Minimize z = 24x + 36y

Subject to

`x + 2y >= 10`

`x + 2y >= 12 or x + y >= 6`

`x + y >=  8`

X + 2y = 10 x + y = 6 3x + 1y = 8
`x/10 + y/5 = 1` x + y = 6 `x/(8/3) + y/8 = 1`

 

Corner points Objective function
z= 24x + 36y
A(10,0) `z= 24xx 10 + 36 xx 0 = 240`
B(2,4) `z = 24 xx 2 + 36 xx 4 = 192`
C(1,5) `z = 24 xx 1 + 36 xx 5 = 204`
D(0,8) `z = 0 xx 24 + 36 xx 8 = 288`

Minimum value of z is 192 at B (2, 4)

  Is there an error in this question or solution?
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