Question

A dietician wishes to mix two types of foods in such away that the vitamin contents of the mixture contains at least $8$ units of vitamin A and $10$ units of vitamin C. Food I contains $2$ units/kg of vitamin A and $1$ unit/kg of vitamin C while Food II contains $1$ unit/kg of vitamin A and $2$ units/kg of vitamin $1$ unit/kg of vitamin C. It costs $Rs.5$ per kg to purchase food I and $Rs.7$ per kg to purchase Food II. Determine the maximum cost of such a mixture. Formulate the above as a LPP and solve it graphically.

Answer 1

Suppose the dietician mixes $x$ kg of food $I$ and $y$ kg of food $II.$

The first condition of the mixture containing $8$ units of vitamin A is given by $2x+y\ge 8$

Also, atleast $10$ units of vitamin C is given by $x+2y\ge 10$

The total cost of the mixture is given by $5x+7y$

So, our problem becomes to maximise $5x+7y$, subject to the conditions $2x+y\ge 8...\left(1\right)$ and $x+2y\ge 10...\left(2\right)$

Equation (1) multiplied by $2$ gives $4x+2y\ge 16$ and when subtracted with equation (2), we get $3x\ge 6$ or $x\ge 2$

When substituted in equation (1), we obtain $y\ge 4$

The minimum cost of such a mixture thus becomes $10+28=Rs.38.$

However, the maximum cost can go upto infinity.

Formation of LPP

To Maximize:	$5x+7y$
Constraints:	$2x+y\ge 8$
	$x+2y\ge 10$
	$x\ge 0,y\ge 0$

Verifying values at corner points:

Corner Points	$(0,8)$	$(2,4)$	$(10,0)$
Value of $5x+7y$	$56$	$38$	$50$

Minimum at $(2,4)$, and It can attain infinity as it's maximum value.