Exercise 9 Page 160

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 $y=a,$ 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 $y=\text{-}x+11.$

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

Inequality Symbol	Region Shaded	Solid/Dashed
$<$	Below	Dashed
$\le$	Below	Solid
$>$	Above	Dashed
$\ge$	Above	Solid

Since all our lines should be solid we will use $\le$ or $\ge ,$ 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 $P=Ax+By,$ where $A$ and $B$ are real numbers. We can choose arbitrary values for $A$ and $B.$ Let $A=2$ and $B=3.$ 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	$2x+3y$	$P=2x+3y$
$(1,2)$	$2(1)+3(3)$	$11$
$(4,5)$	$2(4)+3(5)$	$23$
$(6,5)$	$2(6)+3(5)$	$27$
$(9,2)$	$2(9)+3(2)$	$24$

We can see that the minimum $P\text{-}$value, $11,$ occurs at the point $(1,2),$ and the maximum $P\text{-}$value, $24,$ occurs at the point $(9,2).$