Question

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per
bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units
of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional
element A, 11.25 units of element B, and 3 units of element C. The minimum
requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively.
Determine the number of bags of each brand which should be mixed in order to
produce a mixture having a minimum cost per bag? What is the minimum cost of
the mixture per bag?

Answer 1

Step-by-step explanation:

Let's assume that mixture contains X bags of brands P and Y bags of brands Q.

Cost of 1 bag of brands P =250 Rs

Cost of 1 bag of brands Q =200 Rs

So, Total cost (C) =250X+200Y Rs ...(1)

Now, brands P contains 3 units and brands Q contains 1.5 units of nutritional element A.

So, Total nutritional element A =3X+1.5Y units

Since, minimum requirement of nutritional element A is 18 units.

∴3X+1.5Y≥18

⇒2X+Y≥12 ...(2) (After dividing by 1.5 both side)

Again, brands P contains 2.5 units and brands Q contains 11.25 units of nutritional element B.

So, Total nutritional element B =2.5X+11.25Y units

Since, minimum requirement of nutritional element B is 45 units.

∴2.5X+11.25Y≥45

⇒2X+9Y≥36 ...(3) (After dividing by 1.25 both side)

Also, brands P contains 2 units and brands Q contains 3 units of nutritional element C.

So, Total nutritional element C =2X+3Y units

Since, minimum requirement of nutritional element C is 24 units.

∴2X+3Y≥24 ...(4)

Since, amount of brands can never be negative.

So, X≥0,Y≥0 ...(5)

We have to minimise the cost of mixture given in equation (1) subject to the constraints given in (2),(3), (4) and (5).

After plotting all the constraints, we get the feasible region as shown in the image.

Corner points Value of Z=250X+200Y

A (0,12) 2400

B (3,6) 1950 (minimum)

C (9,2) 2650

D (18,0) 4500

Now, since region is unbounded. Hence we need to confirm that minimum value obtained through corner points is true or not.

Now, plot the region Z<1950 to check if there exist some points in feasible region where value can be less than 1950.

⇒250X+200Y<1950

⇒5X+4Y<39.

Since there is no common points between the feasible region and the region which contains Z<1950 (See the image). So 1950 is the minimum cost.

I hope this will help you :)

Answer 2

Answer:

farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per

bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units

of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional

element A, 11.25 units of element B, and 3 units of element C. The minimum

requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively.

Determine the number of bags of each brand which should be mixed in order to

produce a mixture having a minimum cost per bag? What is the minimum cost of

the mixture per bag?