a rubber ball has the property that on any bounce it returns to 1/3 of the height from which it just fell. Suppose the ball is dropped from 108 ft How far has the ball traveled by the fourth bounce?

To find out how far the ball has traveled by the fourth bounce, we need to calculate the total distance covered during each individual bounce.

Let's break down the problem step-by-step:

1. Bounce 1:
- The ball is dropped from 108 ft.
- It travels a distance of 108 ft on the first bounce.

2. Bounce 2:
- The ball reaches a height of 1/3 * 108 ft after the first bounce, which is equal to 36 ft.
- It then falls back down to the ground from this new height of 36 ft.
- It travels a distance of 36 ft on the second bounce.

3. Bounce 3:
- The ball reaches a height of 1/3 * 36 ft after the second bounce, which is equal to 12 ft.
- It falls back down to the ground from this new height of 12 ft.
- It travels a distance of 12 ft on the third bounce.

4. Bounce 4:
- The ball reaches a height of 1/3 * 12 ft after the third bounce, which is equal to 4 ft.
- It falls back down to the ground from this new height of 4 ft.
- It travels a distance of 4 ft on the fourth bounce.

Now, let's calculate the total distance traveled by summing up the distances covered during each individual bounce:

108 ft (bounce 1) + 36 ft (bounce 2) + 12 ft (bounce 3) + 4 ft (bounce 4) = 160 ft

Therefore, the ball has traveled a total distance of 160 ft by the fourth bounce.

To find out how far the ball has traveled by the fourth bounce, we need to understand the pattern of its bounces.

Let's break down the problem step by step:

1. On the first bounce, the ball reaches a height of 1/3 * 108 ft = 36 ft. This means it has traveled 108 ft - 36 ft = 72 ft during the first bounce.

2. On the second bounce, the ball reaches a height of 1/3 * 36 ft = 12 ft. The total distance traveled by the ball after two bounces is 108 ft (first drop) + 72 ft (first bounce) + 12 ft (second bounce) = 192 ft.

3. On the third bounce, the ball reaches a height of 1/3 * 12 ft = 4 ft. The total distance traveled by the ball after three bounces is 108 ft (first drop) + 72 ft (first bounce) + 12 ft (second bounce) + 4 ft (third bounce) = 196 ft.

4. Finally, on the fourth bounce, the ball reaches a height of 1/3 * 4 ft = 4/3 ft. The total distance traveled by the ball after the fourth bounce is 108 ft (first drop) + 72 ft (first bounce) + 12 ft (second bounce) + 4 ft (third bounce) + 4/3 ft (fourth bounce) ≈ 196 ft + 4/3 ft ≈ 196.33 ft.

Therefore, the ball has traveled approximately 196.33 ft by the fourth bounce.

end of first bounce it dropped 108 ft

during the second bounce it went up and down
(1/3) of 108 or 2(1/3)(108) = 72
during the third bounce it went
2(1/3)(72) = 48
during the fourth bounce it went
2(1/3)(48) = 32

so after 4 bounces it went
108+72+48+32 = 260

make a sketch, label the bounces and see why I only took half of the first bounce cylcle