|
Chapter 3: Applications of Differentiation
Section 3.8: Optimization
|
Example 3.8.2
|
|
|
Show that among all rectangles with a fixed area, the square has the minimum perimeter.
|
|
|
|
|
Solution
|
|
|
|
Analysis
|
|
|
•
|
Figure 3.8.2(a) shows a labeled rectangle whose perimeter is , and whose area is fixed at .
|
|
•
|
The constraint equation is .
|
|
•
|
Solve the constraint equation for, say, and write the perimeter 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 3.8.2(a) Labeled diagram of a rectangle
|
|
|
|
|
|
|
|
|
Computation
|
|
|
Define the objective function
|
|
•
|
Control-drag
|
|
•
|
Context Panel: Assign Function
|
|
|
|
Find critical numbers
|
|
•
|
Write the equation for critical numbers.
Press the Enter key.
|
|
•
|
Context Panel: Solve≻Obtain Solutions for≻
|
|
|
|
Second-Derivative test
|
|
•
|
Write
Context Panel: Evaluate and Display Inline
|
|
=
|
|
|
Of the two critical numbers found, namely, , only the positive root is meaningful since is a dimension. The purist would claim that in addition to the constraint equation , there are additional constraints, namely, that the variables must be nonnegative.
Since , the perimeter is a minimum at , this second equality being determined from the constraint .
|
|
|
<< Previous Example 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
|
