|
Chapter 3: Applications of Differentiation
Section 3.8: Optimization
|
Example 3.8.1
|
|
|
Of all rectangles with perimeter 100, find the one with maximal area.
|
|
|
|
|
Solution
|
|
|
|
Analysis
|
|
|
•
|
Figure 3.8.1(a) shows a labeled rectangle whose area is , and whose perimeter is .
|
|
•
|
The constraint equation is .
|
|
•
|
Solve the constraint equation for, say, and write the area of the rectangle as .
|
|
•
|
Maximize the objective function
|
|
|
>
|
p1:=plottools[rectangle]([1,4],[5,1],style=line):
p2:=plots:-textplot({[3,.7,typeset(w)],[.7,2.5,typeset(h)]},font=[Lucinda,18]):
plots:-display(p1,p2,scaling=constrained, axes=none);
|
|
|
Figure 3.8.1(a) Labeled diagram of a rectangle
|
|
|
|
|
|
|
|
|
Graphical Solution
|
|
|
•
|
Figure 3.8.1(b) is a graph of the objective function . The curve is a parabola, and the vertex of the parabola is the absolute maximum for the function .
|
|
•
|
Using the Context Panel for the graph, set Probe Info to Nearest datum, and trace the graph with the cross-hair form of the probe.
|
|
•
|
Set the cross-hair on the vertex. Again in the Context Panel for the graph, in the Probe Info option, select Copy data.
|
|
•
|
Paste the contents of the Clipboard into the workspace:
|
|
•
|
|
|
|
|
|
Figure 3.8.1(b) Graph of
|
|
|
|
|
|
The density of pixels on the computer screen determines how accurately a graph can be "read." From the graphical approximation, it might be conjectured that the optimal dimensions of the rectangle are , for a maximal area of 625.
|
|
|
Numeric Solution
|
|
|
Numeric solution via the Context Panel
|
|
•
|
Form a sequence of the objective function and the constraint equation.
|
|
•
|
Context Panel: Optimization≻Maximize (local)
|
|
|
|
|
|
The return consists of a list with two objects. The first object is the optimal value of the objective function; the second, a list of the parameter values giving this extreme value.
|
Numeric solution via the Optimization Assistant
|
|
•
|
Form a sequence of the objective function and the constraint equation.
|
|
•
|
Context Panel:
Optimization≻Optimization Assistant
(The Optimization Assistant launches with the objective function and constraint equation inserted into the appropriate fields.)
|
|
•
|
In the Options section, select Maximize
|
|
•
|
Click the Solve button. (See Figure 3.8.1(c).)
|
|
•
|
Click the Quit button to exit the Optimization Assistant and have it write the Solution to the underlying worksheet.
|
|
•
|
Click
to launch the Optimization Assistant with the data embedded.
|
|
|
|
|
Figure 3.8.1(c) Solution by Optimization Assistant
|
|
|
|
|
|
|
|
|
Algebraic Solution
|
|
|
•
|
Control-drag
|
|
•
|
Context Panel: Assign Function
|
|
|
|
•
|
Type and press the Enter key.
|
|
•
|
Context Panel: Expand≻Expand
|
|
•
|
Context Panel: Complete Square≻
|
|
|
|
|
Upon completing the square in , the equation describes a parabola in vertex form. The coordinates of the vertex can be read from the equation: . Hence, the width of the rectangle is 25, its corresponding height is , and the maximal area is 625. The rectangle with fixed perimeter and maximal area is a square.
|
|
|
Analytic Solution
|
|
|
•
|
Write the equation for critical numbers.
Press the Enter key.
|
|
•
|
Context Panel: Solve≻Solve
|
|
|
|
•
|
Determine the maximal area.
|
|
=
|
|
|
If , then the constraint equation determines that , so the rectangle whose perimeter is fixed at 100, and whose area is a maximum, is a square.
The purist might demand that the Second-Derivative test be applied via the calculation , but the pragmatist will insist that Figure 3.8.1(b) is sufficient.
|
|
|
<< Previous Section Section 3.8
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
|
