Figure 3.8.3(a) shows a labeled rectangle subdivided into three equal-area rectangles. The expensive outer fencing is drawn in blue; the cheaper inner fencing, in red.

The objective function is the cost, given by
$C\=5\left(6 u\+2 v\right)\+\frac{7}{2}\left(2 v\right)$, in dollars.

The constraint is $3 u v\=43560$, so that the area enclosed is one acre. Of course, there are the implied constraints that $u$ and $v$ are nonnegative.

Solve the constraint for, say, $v\=14520\/u$ and write the objective function to be minimized as $F\left(x\right)\=30 x\+17 \left(14520\/x\right)$.

Figure 3.8.3(b) is a graph of the objective function $F\left(x\right)\=30 x\+17 \left(14520\/x\right)$.

Given the scale of the graph, a graphical estimate of the minimum point will be difficult; the critical number seems to be approximately 100, and the minimum cost, slightly more than $5,000.

Of course, a more accurate approximation might be obtained if the scale of the graph were modified. That is left to the reader's discretion.

Control-drag a sequence of the objective function and the constraint equation.

Context Panel:
Optimization≻Optimization Assistant

To the right of "Iteration Limit," change "default" to 200.

Press the Solve button to obtain the solution displayed in Figure 3.8.3(c).

Press the Quit button to write the solution to the underlying worksheet.

Click    to launch the Optimization Assistant with the data embedded.

Control-drag $F\left(x\right)\=\dots$
Context Panel: Assign Function

Write the equation for the critical number.
Press the Enter key.

Context Panel: Solve≻Solve

Evaluate $F″$ at the positive critical number.
Context Panel: Evaluate and Display Inline

Evaluate the cost function:
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻10 (digits)

Select the positive critical number.
Context Panel: Approximate≻5 (digits)

Evaluate $v\=14520\/u$ with $u\=22\sqrt{17}$:
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)