Question

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes that can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Answer 1

Let there be x cakes of first kind and y cakes of second kind.

Therefore, x≥0 and y≥0 The given information can be compiled in a table as follows.

Flour(g) Flour(g) Cakes of first kind,x Cakes of second kind,y Availability 200 25 100 50 5000 1000
∴200x+100y≤5000 ⇒2x+y≤50 25x+50y≤1000 ⇒x+2y≤40
Total numbers of cakes,Z,that can be made are, Z=x+y The mathematical formulation of the given problem is

Maximize
Z=x+y...(1)

subject to the constraints,
2x+y≤50...(2)
x+2y≤40...(3)
x,y≥0...(4)

The feasible region determined by the system of constraints is as follows.

Thus the maximum numbers of cakes that can be made is 30(20 of one kind and 10 of the other kind).