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Book Description:
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.
Author:
Sam Masih, Albany State University
E-mail:
smasih@asurams.edu
Topics:
Maxima, Minima, Critical Points, Differentiation, Graphing,
Geometry, Area and Volume calculations
Number of Pages: 9
Animation: Yes
Grade Level: 13-15
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to enter this 9-page Microworld.