Pearson Algebra 2 Common Core, 2011
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Pearson Algebra 2 Common Core, 2011 View details
4. Linear Programming
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Exercise 9 Page 160

What characterizes a trapezoid? Start by thinking about what your boundary lines should be for them to trace a trapezoid.

Example System of Inequalities:

Objective Function
Table:

Vertex
Practice makes perfect

We are asked to find a system of constraints whose graph determines a trapezoid, and then to write an objective function and evaluate it at each vertex of the trapezoid. We will start by finding the system of constraints.

Writing the Constraints

Since a trapezoid has one pair of parallel sides, we can start by tracing two parallel lines. We can do this by tracing two different horizontal lines. These are of the form where is a constant. We can try, for example, and

Now, to do the other sides we can use an equation with a positive slope and one with a negative slope — for instance, and

Now that we found the boundary lines, we just need to choose the appropriate inequality symbol.

Inequality Symbol Region Shaded Solid/Dashed
Below Dashed
Below Solid
Above Dashed
Above Solid
Since all our lines should be solid we will use or making sure they shade over the central trapezoid.
We found our system of constraints.
Notice that we could have chosen different equations for the boundary lines, and this system is only one example solution. There are infinitely many possible solutions.

Writing and Evaluating the Objective Function

Now we need to write an objective function. For this we need to chose any linear function in the form where and are real numbers. We can choose arbitrary values for and Let and
We now proceed by locating the vertices of the feasible zone.

Finally, we proceed to evaluate the objective function at all of the vertices to find the minimum and maximum values.

Vertex

We can see that the minimum value, occurs at the point and the maximum value, occurs at the point