a ball is dropped from a height of 5m. after each bounce, the ball rises to 45% of its previous height. what the total vertical distance that the ball has travelled after it has hit the ground for for the 6th time?

what are the formulas? because the ball bounces twice each time

5 [1 + 2*(0.45) +2*(0.45^2) + 2(0.45)^3 + 2*(0.45)^4 + 2*(0.45)^5] = __

2.606

To find the total vertical distance that the ball has traveled after hitting the ground for the sixth time, we need to calculate the sum of the distances traveled during each bounce.

Given:
Initial height (h) = 5m
The ball rises to 45% (0.45) of its previous height after each bounce.
The ball bounces twice each time it hits the ground.

Formulas:
1. Distance traveled during each bounce = 2 * previous height * 0.45
2. Total vertical distance traveled = sum of distances traveled during each bounce

Step-by-step solution:
1. Calculate the distance traveled during each bounce for the first six bounces:
- First bounce: 2 * 5m * 0.45 = 4.5m
- Second bounce: 2 * 4.5m * 0.45 = 4.05m
- Third bounce: 2 * 4.05m * 0.45 = 3.645m
- Fourth bounce: 2 * 3.645m * 0.45 = 3.2805m
- Fifth bounce: 2 * 3.2805m * 0.45 = 2.95245m
- Sixth bounce: 2 * 2.95245m * 0.45 = 2.657205m

2. Calculate the total vertical distance traveled:
Total distance = 5m + 4.5m + 4.05m + 3.645m + 3.2805m + 2.95245m + 2.657205m

Therefore, the total vertical distance traveled by the ball after hitting the ground for the sixth time is approximately 26.08516m.

To find the total vertical distance traveled by the ball after it has hit the ground for the 6th time, we need to consider the distance covered during both the upward and downward motions of each bounce.

Let's break down the problem step by step:

1. First, we need to determine the total distance traveled during each bounce. Since the ball rises to 45% of its previous height after each bounce, it covers a distance of 0.45 times the previous height when it rises.

2. The ball starts by falling from a height of 5m, so during the first bounce, it will travel a distance of 5m when it falls and an additional distance of 0.45 * 5m when it rises.

3. For subsequent bounces, the distance traveled during the upward motion reduces each time, but we can calculate it using the same formula. After the first bounce, the ball reaches a maximum height of 0.45 * 5m = 2.25m. From this height, it falls and then rises to 0.45 * 2.25m = 1.0125m. This process continues for each bounce.

4. To find the total distance traveled after the 6th bounce, we need to sum up the distance covered during each bounce.

Now let's calculate the distances for each bounce:

Bounce 1:
Distance covered when falling = 5m
Distance covered when rising = 0.45 * 5m = 2.25m

Bounce 2:
Distance covered when falling = 2.25m
Distance covered when rising = 0.45 * 2.25m = 1.0125m

Bounce 3:
Distance covered when falling = 1.0125m
Distance covered when rising = 0.45 * 1.0125m = 0.455625m

Bounce 4:
Distance covered when falling = 0.455625m
Distance covered when rising = 0.45 * 0.455625m = 0.20553125m

Bounce 5:
Distance covered when falling = 0.20553125m
Distance covered when rising = 0.45 * 0.20553125m = 0.09298806m

Bounce 6:
Distance covered when falling = 0.09298806m
Distance covered when rising = 0.45 * 0.09298806m = 0.04184463m

Finally, let's calculate the total distance traveled by summing all the distances covered during each bounce:

Total distance traveled after 6 bounces = Distance of Bounce 1 + Distance of Bounce 2 + Distance of Bounce 3 + Distance of Bounce 4 + Distance of Bounce 5 + Distance of Bounce 6

Total distance traveled after 6 bounces = 5m + 2.25m + 1.0125m + 0.455625m + 0.20553125m + 0.09298806m + 0.04184463m

Total distance traveled after 6 bounces = 9.05848894m

Therefore, the total vertical distance traveled by the ball after it has hit the ground for the 6th time is approximately 9.06 meters.