Page 72 - Mathematics_Form_3

Linear programming
7. A patient needs 5 mg, 10 mg, and
15 mg of vitamins A, B, and C per
day, respectively, from mangoes
and oranges. A mango has 0.5 mg
of vitamin A, 2 mg of vitamin B,
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and 3 mg of vitamin C. An orange
has 1 mg of vitamin A, 3 mg of
vitamin B, and 3 mg vitamin C.
The costs of the mango and the
orange are Tshs 400 and Tshs 150,
respectively. Formulate a cost
minimization linear programming
problem.
8. A dietitian wants to blend two types
of ingredients, I and I , so that the
2
1
mixture’s vitamin content includes
at least 6 units of vitamin A and 8
units of vitamin B. Ingredient I
1
contains 2 units/kg of vitamin A
and 3 units/kg of vitamin B, while
ingredient I contains 3 units/kg
2
of vitamin A and 4 units/kg of
vitamin B. Ingredient I and I cost
2
1
Tshs 8,000 per kilogram and Tshs
7,500 per kilogram, respectively.
Formulate a linear programming
problem to minimize the cost of
the mixture.
9. A manufacturer has 90, 80, and 60
running metres of plywood, pine,
and birch, respectively. An item
of product A requires 2, 2, and 4
running metres of plywood, pine,
and birch, respectively, and an item
of product B requires 4, 5, and 1
running metres of plywood, pine,
and birch, respectively. If an item
of product A is sold at Tshs 10,000
Student\s Book Form Three 65
MATHEMATIC F3 SB.indd 65
MATHEMATIC F3 SB.indd 65
and that of product B is sold at
Tshs 13,000. Formulate a linear
programming problem for finding
the optimal sales of the product.
10. Petro had 250,000 Tanzanian
shillings to buy oranges and
mangoes. An orange costs Tshs 100
while a mango costs Tshs 500. If he
needed to buy at least twice as many
mangoes as oranges, formulate a
linear programming problem to
represent this information.
Graphical solutions
The constraints of the linear programming
problem can be solved graphically.
The solution is obtained by treating
the inequalities as linear equations but
the set of inequalities will be satisfied
by the obtained region. The following
terminologies are useful when graphing
linear programming problems.
Feasible region
A feasible region of a linear programming
problem is the set of all possible values
of the decision variables which satisfies
all the constraints of the problem. If the
feasible region is enclosed by a polygon,
it is said to be bounded, otherwise it is
unbounded. Mathematics for Secondary Schools
Optimal solution
An optimal solution is any solution in
the feasible region that gives the optimal
value of the objective function. It is the
largest value of the objective function for
a maximization problem and the smallest
value for a minimization problem.
18/09/2025 09:59:08
18/09/2025 09:59:08