12. A rubber ball falls from the roof of a house of height $10 \mathrm{~m}$. Each time it hits the ground, it rebounds to four-fifths of the height it fell from previously. How many times will the ball appear in front of a $1 \mathrm{~m}$ high window whose bottom edge is $5 \mathrm{~m}$ above the ground?
A 1
B 2
C 4
D 6
E 8
Solution
D

As the ball falls from a height of $10 \mathrm{~m}$, it will appear in front of the window on its way down. As shown in the diagram, after hitting the ground, it will rebound to a height of $8 \mathrm{~m}$, which is above the upper edge of the window, and so will appear in front of the window on its way up and on its way down. After hitting the ground a second time, it will rebound to a height of $6.4 \mathrm{~m}$, which is also above the upper edge of the window, and so will again appear in front of the window on its way up and on the way down. After hitting the ground a third time, it will rise to a height of $5.12 \mathrm{~m}$ and so will appear in front of the window before disappearing from the direction it came. After the fourth bounce, the ball only reaches a height of $4.096 \mathrm{~m}$ and so does not appear in front of the window. When you count up the number of times the ball appeared in front of the window, you find it has appeared six times.