Microworld: Optimize
: (All in One)
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to visit the Microworld.
Author:
Samuel
Masih
This
9-page Microworld, which is based on the earlier Mathwright WorkBook: The
Classic Box Problem, by Charles and Rosanne Hoffman (the former of Villanova
University), takes as its theme: The visualization of maximization problems.
It presents a sequence of problems that are masterfully chosen to help the
reader see what optimization means in the context of a lively and interactive
environment.
In
every case, the author carefully orchestrates three concepts: The notion of
a principle parameter that determines a changing quantity, the correspondence
between the graph of the quantity to be optimized as function of this parameter,
and a geometric visualization of the object that is determined by the parameter,
and finally, the role of the derivative of the quantity in determining the
optimal parameter value.
The
starting point in the series is the familiar Open Box of Maximum Volume
problem: "From a rectangular sheet of cardboard a square of sides x is
cut from each corner of the sheet. The sheet is then folded to get a box Find
x to get a box of maximum volume." In this presentation, the reader experimentally
chooses arbitrary values for "x" and sees 2- and 3-dimensional representations
of the resulting boxes, along with numeric and graphical representations of
its dimensions and volume. At any time, the reader may draw the graph of the
volume function as function of x, and then select points along the graph to
see the corresponding boxes themselves in both 2 and 3 dimensions. Thus, the
optimization procedure "comes to life" in the reader's eyes in a
colorful and exciting way.
The
Microworld then moves on to explore the problem of finding the largest area
of a rectangle that can be inscribed within a given triangle. The reader constructs
the triangle as she likes, and examines in an interactive process, the various
rectangles, their dimensions and areas, as they are determined by a single
parameter. Those rectangles are drawn within the triangle, and their areas
are simultaneously plotted as function of this parameter. The area function
may be drawn at any time, and then the reader may select points on the graph
and see how they correspond with the rectangles associated to them.
As
a final topic, the calculation of volume subject to constraints is revisited
in the Postman's Problem: "Find the width and the height of a box of
given length accepted by the post office for mail. It is assumed that the
post office requires that the girth of the box not exceed 108 inches. Girth
= 2*w+2*h+l = 108, where w = width, h = height, and l = given length.
The
Microworld explores these themes in 8 interactive pages + a Table of Contents.
Return to the listing of MathwrightWeb Microworlds
| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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