Download worksheet for classroom use.
The worksheet contains 5 problems where the inequalities are
given and students are asked to find the feasible regions and
label the coordinates of the vertices of the polygons
representing the feasible regions. The worksheet is
appropriate for use with, or without, the graphing calculator.
Discussion of Linear Programming to precede worksheet:
Linear Programming ("Planning") is an application of
mathematics to such fields as business, industry, social
science, economics, and engineering. The process is
used to establish feasible regions and locate maximum and minimum values
which can take place under certain given conditions.
Linear programming was developed as a discipline in the 1940's by George
Dantzig, John von Neumann, and Leonid Kantorovich.
Let's start our investigation into Linear Programming by establishing
feasible regions. These feasible regions are simply the solutions to
systems of inequalities, such as those we have been studying. Feasible
regions are all locations that represent "feasible" (viable, possible,
correct) solutions to the set of inequalities.
Establish a feasible region for the following set
Determine the coordinates of the vertices of
the polygon formed by the feasible region.
The feasible region is shaded in yellow. The coordinates
of the polygon are (0,8), (0,1), (7,1).
Once the feasible region has been established, linear
programming then examines the function which is to be maximized or
minimized within this feasible region.
Let's assume that the function to be examined for Example 1 will be
Each vertex of the polygon is tested in the function.
2(0) + 3(8) = 0 + 24 = 24
2(0) + 3(1) = 0 + 3 = 3
2(7) + 3(1) = 14 + 3 = 17
A maximum value is at (0,8) and a minimum is at (0,1).
For our study of Linear
Programming, we will be limiting our investigation to the locations of
feasible regions and the vertices of the polygons formed.
Find a feasible region to represent this situation.
Student Council is making colored armbands for
the football team for an upcoming game. The school's
colors are orange and black. After meeting with students
and teachers, the following conditions were established:
1. The Council must make at least one black armband but
not more than 4 black armbands since the black armbands might be
seen as representing defeat.
Council must make no more than 8 orange armbands.
3. Also, the number of black armbands
should not exceed the number of orange armbands.
Let x = black armbands
y = orange armbands
1. x > 1
and x < 4
2. y < 8
3. x < y
It will be assumed that these numbers are not negative at any
The feasible region is shaded in yellow.
of the polygon are (1,8), (1,1), (4,4) and (4,8).