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### Pool Table Algebra

```
Date: 10/21/98 at 22:20:50
From: Roland
Subject: Algebra

The y-axis, the x-axis, the line x = 6, and the line y = 12 determine
the four sides of a 6 by 12 rectangle in the first quadrant of the xy
plane. Imagine that this rectangle is a pool table. There are pockets
at the four corners and at the points (0,6) and (6,6) in the middle of
each of the longer sides. When a ball bounces off one of the sides of
the table, it obeys the "pool rule," the slope of the path after the
bounce is the negative of the slope before the bounce.

Your pool ball is at (3,8). You hit it toward the y-axis, along the
line with slope 2.

a. Where does it hit the y-axis?

b. If the ball is hit hard enough, where does it hit the side of the
table next? And after that? And after that?

c. Does it ultimately return to (3,8)? Would it do this if the slope
had been different from 2? What is special about the slope 2 for
this table?
```

```
Date: 10/22/98 at 12:34:14
From: Doctor Peterson
Subject: Re: Algebra

Hi, Roland. This problem is obviously meant to give you a chance to
explore a bit and see how the problem works before you get to the final
answer, so I'll just get you started. Here's a picture of the table and
the ball:

|           |
12+-----------+---
|           |
|           |
|           |
|     *     |
|           |
|           |
|           |
|           |
|           |
|           |
|           |
0+-----------+---
0           6

If you hit the ball toward the y axis with a slope of 2, it looks like
this:

|           |
12+-----------+---
|           |
|           |
|           |
|     *     |
|    /      |
|   /       |
|  /        |
| /         |
|/          |
*           |
|           |
0+-----------+---
0           6

To find where it hits, you have to write the equation of the line and
find its y-intercept. (There are lots of other ways to do this, such
as geometry, but I suspect you are supposed to work with lines and
slopes.) Once you find that, you can write the equation of the next
line the ball will roll on, using that intercept and the new slope, -2:

|           |
12+-----------+---
|           |
|           |
|           |
|     *     |
|    /      |
|   /       |
|  /        |
| /         |
|/          |
*           |
|\          |
0+-*---------+---
0           6

Keep working this way. For the next two lines you will have to find
where it intersects the lines x = 6 and y = 12, respectively, which
will be just a little harder. Each time, check whether (3,8) is on the
line, to see whether you will get back to the starting point. You may
want to draw it on graph paper, which will help you see what's
happening. Have fun with it!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Linear Equations
High School Puzzles

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