A ball is dropped from a height of 80ft.The elasticity of this ball is such that it rebounds three-fourths of the distance it has fallen. How high does the ball rebound on the fifth bounce? Find a formula for the height the ball rebounds on the nth bounce?

To find the height the ball rebounds on the fifth bounce, we can use the given information that the ball rebounds three-fourths of the distance it has fallen.

Let's break down the problem into individual bounces:

1st bounce:
- The ball is dropped from a height of 80ft.
- It rebounds three-fourths of the distance it has fallen: 3/4 * 80ft = 60ft.

2nd bounce:
- The ball is dropped from a height of 60ft after the first bounce.
- It rebounds three-fourths of the distance it has fallen: 3/4 * 60ft = 45ft.

3rd bounce:
- The ball is dropped from a height of 45ft after the second bounce.
- It rebounds three-fourths of the distance it has fallen: 3/4 * 45ft = 33.75ft.

4th bounce:
- The ball is dropped from a height of 33.75ft after the third bounce.
- It rebounds three-fourths of the distance it has fallen: 3/4 * 33.75ft = 25.31ft.

5th bounce:
- The ball is dropped from a height of 25.31ft after the fourth bounce.
- It rebounds three-fourths of the distance it has fallen: 3/4 * 25.31ft = 18.98ft.

To find a formula for the height the ball rebounds on the nth bounce, we can observe that the height of each bounce is decreasing by three-fourths (3/4) of the previous bounce's height.

Therefore, the formula for the height the ball rebounds on the nth bounce is: H(n) = (3/4)^n * 80ft, where H(n) represents the height on the nth bounce and 80ft is the initial height the ball is dropped from.

To find the height the ball rebounds on the fifth bounce, we need to calculate the distance it travels each time it bounces. Let's break down the problem step by step:

1. First, let's determine the distance the ball travels on the first bounce. We know that the ball is dropped from 80 ft, so it falls 80 ft initially. After hitting the ground, it rebounds three-fourths of the distance it has fallen, which means it bounces back up to 3/4 * 80 ft = 60 ft.

2. On the second bounce, the ball is now at a height of 60 ft and falls from there, covering a distance of 60 ft. It rebounds three-fourths of this distance, resulting in a rebound height of 3/4 * 60 ft = 45 ft.

3. On the third bounce, the ball starts from 45 ft high, falls, and rebounds three-fourths of the distance it has fallen. So it will rebound to 3/4 * 45 ft = 33.75 ft.

4. Similarly, on the fourth bounce, the ball begins from a height of 33.75 ft, falls, and rebounds three-fourths of the distance, resulting in a rebound height of 3/4 * 33.75 ft = 25.3125 ft.

5. Now, for the fifth bounce, we follow the same pattern. The ball falls from 25.3125 ft, and its rebound height would be 3/4 * 25.3125 ft.

To find the formula for the height the ball rebounds on the nth bounce, we can write it as:

Height(n) = (3/4) * Height(n-1)

Where Height(n-1) is the height of the (n-1)th bounce.

By repeatedly applying this formula, you can find the height of rebound for any bounce, including the fifth bounce.

bounce height

1: 80 * 3/4 = 80(3/4)^1
2: 80 * 3/4 * 3/4 = 80(3/4)^2
...