Lizette drops a ball from a height

of 256 feet. The peak height after each bounce is 75% the previous height. How
much vertical distance does the ball travel from the time she drops it until it touches the ground after its
fourth
bounce

To find the total vertical distance traveled by the ball until it touches the ground after its fourth bounce, we need to calculate the distance traveled during each bounce and add them up.

Let's break down the problem step by step:

Step 1: Calculate the distance traveled during each bounce:
The initial drop is considered the first bounce, so we have a total of four bounces.

For the first bounce, the ball drops from a height of 256 feet. The peak height after the first bounce is 75% of the previous height.

Height after the first bounce = 256 * 0.75 = 192 feet

For the second bounce, the peak height is 75% of the height obtained after the first bounce.

Height after the second bounce = 192 * 0.75 = 144 feet

For the third bounce, the peak height is calculated similarly:

Height after the third bounce = 144 * 0.75 = 108 feet

And for the fourth bounce:

Height after the fourth bounce = 108 * 0.75 = 81 feet

Step 2: Calculate the distance traveled during each bounce:
The distance traveled during each bounce is twice the height because the ball goes up and then comes down.

Distance traveled during the first bounce = 2 * 256 = 512 feet
Distance traveled during the second bounce = 2 * 192 = 384 feet
Distance traveled during the third bounce = 2 * 144 = 288 feet
Distance traveled during the fourth bounce = 2 * 108 = 216 feet

Step 3: Calculate the total vertical distance traveled by adding all four bounces:

Total vertical distance traveled = Distance in the first bounce + Distance in the second bounce + Distance in the third bounce + Distance in the fourth bounce

Total vertical distance traveled = 512 + 384 + 288 + 216 = 1400 feet

Therefore, the ball travels a vertical distance of 1400 feet from the time Lizette drops it until it touches the ground after its fourth bounce.

256 + 2*256*.75 + ... + 2*256*.75^4

= 2*256(1-.75^5)/(1-.75) - 256
= 1306