A ball dropped from a height of one m. Every time it hits the ground, it rebounds to exactly 2/3 of the previous height. What is the total distance the ball has traveled when it hits the ground for the 8th time? (Round to two decimal places.)
To find the total distance traveled when the ball hits the ground for the 8th time, we need to sum up the distances traveled during each bounce.
The distance the ball travels during the first bounce is 1 meter (initial drop height).
For the subsequent bounces, the distance traveled is 2/3 of the previous bounce. So the distances for each bounce are as follows:
1st bounce: 1 meter
2nd bounce: (2/3) * 1 meter = 0.67 meters
3rd bounce: (2/3) * 0.67 meters = 0.44 meters
4th bounce: (2/3) * 0.44 meters = 0.29 meters
5th bounce: (2/3) * 0.29 meters = 0.19 meters
6th bounce: (2/3) * 0.19 meters = 0.13 meters
7th bounce: (2/3) * 0.13 meters = 0.09 meters
8th bounce: (2/3) * 0.09 meters = 0.06 meters
Now we sum up all the distances:
1 + 0.67 + 0.44 + 0.29 + 0.19 + 0.13 + 0.09 + 0.06 = 3.87
Therefore, the total distance the ball has traveled when it hits the ground for the 8th time is 3.87 meters.