A certain ball, when dropped from a height rebounds to one half of the original height.Suppose thos ball is dropped from a point 8ft above the ground, how far has it traveled, counting up and down distance only, when it hits the ground for the eight time?
distance of 1st bounce --- 8
distance of 2nd bounce --- 2(4) = 8
distance of 3rd bounce --- 2(2) = 4
distance of 4th bounce --- 2(1) = 2
distance of 5th bounce --- 2(1/2) = 1
distance of 6th bounce --- 2(1/4) = 1/2
distance of 7th bounce --- 2(1/8) = 1/4
distance of 8th bounce --- 2(1/16)= 1/8
Since there are only 8 terms, we can just add them up
I got a sum of 191/8
If there had been more terms, notice that , not counting the first term, we have a GP with a = 8 and r = 1/2
sum = 8 + sum(7)
= 8 + 8(1 - (1/2)^7)/(1- 1/2)
= 8 + 8(127/128) / (1/2)
= 8 + 8(127/128)(2)
= 191/8 , just as above
To find out how far the ball has traveled, we need to determine the total distance covered during each bounce.
Given that the ball rebounds to half of its original height, we can create the following sequence:
1st drop: 8 ft
1st bounce: (1/2) * 8 ft = 4 ft
2nd drop: 4 ft
2nd bounce: (1/2) * 4 ft = 2 ft
3rd drop: 2 ft
3rd bounce: (1/2) * 2 ft = 1 ft
4th drop: 1 ft
4th bounce: (1/2) * 1 ft = 0.5 ft
The distances covered for each drop and bounce form a geometric sequence with a common ratio of 1/2. The total distance traveled for each bounce can be calculated using the formula:
Total distance = drop distance + (bounce distance * 2)
For each drop and bounce pair, the total distance is equivalent to the drop distance since the bounce distance is added to both the ascending and descending portions.
Now, let's calculate the total distance covered for 8 bounces:
Total distance = 8 ft + 4 ft + 4 ft + 2 ft + 2 ft + 1 ft + 1 ft + 0.5 ft
Total distance = 22.5 ft
Therefore, when the ball hits the ground for the eighth time, it has traveled a total distance of 22.5 ft.
To determine the distance the ball has traveled, we need to calculate the total distance covered during each bounce. Since the ball rebounds to half of its original height, it will reach a maximum height of 4 feet on each bounce.
Considering the first drop from a height of 8 feet, we can break down the ball's movement as follows:
1. Drop from 8 feet to the ground: The ball covers 8 feet.
2. First bounce from the ground to a height of 4 feet: The ball covers 4 feet.
3. First descent from 4 feet to the ground: The ball covers another 4 feet.
4. Second bounce from the ground to a height of 4 feet: The ball covers 4 feet.
5. Second descent from 4 feet to the ground: The ball covers another 4 feet.
6. Third bounce from the ground to a height of 4 feet: The ball covers 4 feet.
7. Third descent from 4 feet to the ground: The ball covers another 4 feet.
8. Fourth bounce from the ground to a height of 4 feet: The ball covers 4 feet.
To calculate the total distance covered when the ball hits the ground for the eighth time, we sum up the distances covered during each action:
8ft + 4ft + 4ft + 4ft + 4ft + 4ft + 4ft + 4ft = 36 feet
Therefore, the ball will travel a total distance of 36 feet when it hits the ground for the eighth time.