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/
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