Cantilevered Blocks
Imagine that we have a supply of toy blocks of uniform density, all of the same
size and weight, each one foot long. We also have a strong table, anchored
securely to the floor. Our job is to arrange a stack of blocks on the table so
that the stack extends out beyond the right edge of the table. When building
the stack, the following rules apply:
- The first block must be placed so that it extends beyond the right edge
of the table.
- Every other block must be placed on top of the previously added block,
and it must extend further right.
Here is the question that we will study in this lesson: How far out beyond
the right edge of the table can the right edge of the topmost block extend?
Let's consider two simple examples to be sure that you understand the problem.
Figure 1: One Block
For the first example,
let's place a single block so that it extends halfway off the table. It is
thus perfectly balanced on top of the table. Unfortunately, no matter how
little beyond the right edge of the first block the second block extends, the
first block will ``lose its balance'' and the pile will collapse. Try this
with a couple of identical books if you don't believe this.
Figure 2: Two Blocks
For the second example,
let's place one block so that one third of it extends off of the table, and a
second block so that one third of it extends beyond the first block. Although
this stack is stable, the bottom block is now perfectly balanced. The next
block that is added at the top of the stack will cause the bottom block to
fall. Again, it may be helpful to do an experiment with books.
In the first example, we succeeded in extending the top block half of a foot
away from the table; in the second example, two thirds of a foot. Can we do
better? If so, how much better?
As we study this question we will:
- see how a computational tool like Maple can be useful in coming up with
an initial model,
- observe the power of Maple's exact integer and rational number
arithmetic, and
- learn how to use built-in functions to do more interesting things with
Maple.
Joseph L. Zachary
Hamlet Project
Department of Computer Science
University of Utah