The y-axis, the x-axis, the line x 5 6, and the line y 5

Chapter 2, Problem 2.7.20

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The y-axis, the x-axis, the line x 5 6, and the line y 5 12 determine the four sides of a 6-by-12 rectangle in the first quadrant (where x . 0 and y . 0) 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. (Hint: It helps to sketch the pool table on a piece of graph paper first.) a. Your pool ball is at (3, 8). You hit it toward the y-axis, along the line with slope 2. i. Where does it hit the y-axis? ii. If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? iii. Show that the ball ultimately returns 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? b. A ball at (3, 8) is hit toward the y-axis and bounces off it at . Does it end up in one of the pockets? If so, what are the coordinates of that pocket? c. Your pool ball is at (2, 9). You want to shoot it into the pocket at (6, 0). Unfortunately, there is another ball at (4, 4.5) that may be in the way. i. Can you shoot directly into the pocket at (6, 0)? ii. You want to get around the other ball by bouncing yours off the y-axis. If you hit the y-axis at (0, 7), do you end up in the pocket? Where do you hit the line x 5 6? iii. If bouncing off the y-axis at (0, 7) didnt work, perhaps there is some point (0, b) on the y-axis from which the ball would bounce into the pocket at (6, 0). Try to find that point.