You drop a ball from a height of 128 meters. Each time it hits the ground, it bounces 50% of its
previous height. How high does the ball go after the ninth time it hits the ground?
To find the height the ball goes after the ninth bounce, we need to calculate the height of each bounce and add them up.
Given:
Initial height: 128 meters
Bouncing height: 50% of previous height
Let's calculate the height after each bounce:
1st bounce: 128 meters * 0.5 = 64 meters
2nd bounce: 64 meters * 0.5 = 32 meters
3rd bounce: 32 meters * 0.5 = 16 meters
4th bounce: 16 meters * 0.5 = 8 meters
5th bounce: 8 meters * 0.5 = 4 meters
6th bounce: 4 meters * 0.5 = 2 meters
7th bounce: 2 meters * 0.5 = 1 meter
8th bounce: 1 meter * 0.5 = 0.5 meters
9th bounce: 0.5 meters * 0.5 = 0.25 meters
After the ninth time it hits the ground, the ball will go up to a height of 0.25 meters.
To determine how high the ball goes after the ninth bounce, we can start by finding the height of each bounce.
Given that each bounce is 50% of the previous height, we can represent this mathematically as:
Height of bounce n = (50/100) * Height of bounce n-1
We can calculate the height after each bounce, starting from the initial height of 128 meters.
First bounce: Height = (50/100) * 128 = 64 meters
Second bounce: Height = (50/100) * 64 = 32 meters
Third bounce: Height = (50/100) * 32 = 16 meters
Fourth bounce: Height = (50/100) * 16 = 8 meters
Fifth bounce: Height = (50/100) * 8 = 4 meters
Sixth bounce: Height = (50/100) * 4 = 2 meters
Seventh bounce: Height = (50/100) * 2 = 1 meter
Eighth bounce: Height = (50/100) * 1 = 0.5 meters
Ninth bounce: Height = (50/100) * 0.5 = 0.25 meters
Therefore, after the ninth bounce, the ball reaches a height of 0.25 meters.
This just another GP problem, with
a = 128
r = 1/2