Page 78 - Mathematics_Form_3

Linear programming

Corner points Values of the objective function f (x, y) = 400000x + 600000y

A (0, 0) f = 400000(0) + 600000(0) = 0

B (9, 0) f = 400000(9) + 600000(0) =3600000
C (7.2, 3.6) f = 400000(7.2) + 600000(3.6) = 5040000
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D (2.4, 7.2) f = 400000(2.4) + 600000(7.2) = 5280000

E (0, 8) f = 400000(0) + 600000(8) = 4800000

The maximum value of the objective function is at point D(2.4, 7.2) which is
Tshs 5,280,000. It is important to note that, the number of products manufactured
cannot be fractional.

Therefore, in order to obtain maximum gross income, the manufacturer should
make 2 units of product A and 7 units of product B.

Example 3.8

John requires 10, 12, and 12 units of chemicals of types A, B, and C, respectively for
his farm. A liquid product contains 5, 2, and 1 units of A, B, and C, respectively per
litre, while a powder product contains 1, 2, and 4 units of A, B, and C, respectively
per carton. If a litre of the liquid costs Tshs 3,000 and a carton of the powder costs
Tshs 2,000, how many units of each should John purchase so as to minimize the
cost but meet the requirements?

Solution
The given information are interpreted and summarized as shown in the following
table.
Product Chemicals Costs

A B C (Tshs)
Liquid 5 2 1 3,000 Mathematics for Secondary Schools
Powder 1 2 4 2,000
Minimum requirements 10 12 12

Let x be the number of litres of the liquid product, and

y be the number of cartons of the powder product to be purchased.
Thus, the objective function is given by,
Minimize f (x, y) = 3000x + 2000y

Student\s Book Form Three 71

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MATHEMATIC F3 SB.indd 71