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What characterizes a trapezoid? Start by thinking about what your boundary lines should be for them to trace a trapezoid.
Example System of Inequalities:
⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧y≤5y≥2y≤x+1y≤-x+11
Objective Function P=2x+3y
Table:
Vertex | P=2x+3y |
---|---|
(1,2) | 11 |
(4,5) | 23 |
(6,5) | 27 |
(9,2) | 24 |
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.
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 y=a, where a is a constant. We can try, for example, y=2 and y=5.
Now, to do the other sides we can use an equation with a positive slope and one with a negative slope — for instance, y=x+1 and y=-x+11.
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 |
Finally, we proceed to evaluate the objective function at all of the vertices to find the minimum and maximum values.
Vertex | 2x+3y | P=2x+3y |
---|---|---|
(1,2) | 2(1)+3(3) | 11 |
(4,5) | 2(4)+3(5) | 23 |
(6,5) | 2(6)+3(5) | 27 |
(9,2) | 2(9)+3(2) | 24 |
We can see that the minimum P-value, 11, occurs at the point (1,2), and the maximum P-value, 24, occurs at the point (9,2).