A bouncy ball is dropped such that the height of its first bounce is 3.75 feet and each successive bounce is 77% of the previous bounce's height. What would be the height of the 12th bounce of the ball?
Well, I hope that bouncy ball isn't afraid of heights! Let's calculate the height of the 12th bounce.
Starting with the first bounce at 3.75 feet, we have to keep multiplying that height by 0.77 to find the subsequent bounces. So, the formula for each bounce is:
height of bounce n = height of bounce (n-1) * 0.77
To find the height of the 12th bounce, we'll start with the height of the first bounce:
height of 1st bounce = 3.75 feet
And then apply the formula 12 times:
height of 2nd bounce = 3.75 * 0.77
height of 3rd bounce = (3.75 * 0.77) * 0.77
...
height of 12th bounce = (3.75 * 0.77)^11
Now, let's calculate that final height:
height of 12th bounce ≈ 0.25 feet
Well, I guess that bouncy ball's got some ups and downs! So, according to my calculations, the height of the 12th bounce would be approximately 0.25 feet.
To find the height of the 12th bounce, we need to find the height at each bounce leading up to the 12th bounce. We are given that the first bounce has a height of 3.75 feet, and each successive bounce is 77% of the previous bounce's height.
Let's calculate the height at each bounce:
1st bounce: 3.75 feet
2nd bounce: 3.75 * 0.77 = 2.8875 feet
3rd bounce: 2.8875 * 0.77 = 2.227925 feet
4th bounce: 2.227925 * 0.77 = 1.71761275 feet
5th bounce: 1.71761275 * 0.77 = 1.3224261725 feet
6th bounce: 1.3224261725 * 0.77 = 1.01977417563 feet
7th bounce: 1.01977417563 * 0.77 = 0.785214619196 feet
8th bounce: 0.785214619196 * 0.77 = 0.60423212812172 feet
9th bounce: 0.60423212812172 * 0.77 = 0.465817326918324 feet
10th bounce: 0.465817326918324 * 0.77 = 0.3582342110901395 feet
11th bounce: 0.3582342110901395 * 0.77 = 0.27596238418660465 feet
Now, we can calculate the height of the 12th bounce:
12th bounce: 0.27596238418660465 * 0.77 = 0.2125150178004372 feet
Therefore, the height of the 12th bounce of the ball would be approximately 0.2125 feet.
To find the height of the 12th bounce of the ball, we can use the given information that each successive bounce is 77% of the previous bounce's height.
First, let's find the height of the 2nd bounce. We can calculate it by multiplying the height of the first bounce (3.75 feet) by 0.77 (77%):
2nd bounce height = 3.75 feet * 0.77 ≈ 2.8875 feet
Next, let's find the height of the 3rd bounce by multiplying the height of the 2nd bounce by 0.77:
3rd bounce height = 2.8875 feet * 0.77 ≈ 2.225575 feet
We can continue this process to find the heights of the 4th, 5th, and subsequent bounces. However, instead of calculating the bounces one at a time, we can use a formula to find the height of any bounce.
The formula to find the height of any bounce (n) is:
bounce height (n) = first bounce height * 0.77^(n-1)
Using this formula, we can find the height of the 12th bounce:
12th bounce height = 3.75 feet * 0.77^(12-1)
Calculating the value:
12th bounce height ≈ 3.75 feet * 0.77^11 ≈ 0.7499 feet
Therefore, the height of the 12th bounce of the ball is approximately 0.7499 feet.