To solve this problem, we need to convert all the measurements to a consistent unit. Let's convert the height to inches since the height of the bounce is given in inches.
64 ft = 64 * 12 inches = 768 inches
Now, we can set up an equation to represent the height of each bounce. Let's use "b" to represent the number of bounces, and "h" to represent the height of each bounce in inches.
The height of each bounce is three-fourths (3/4) the height of the previous bounce. So, we can write the equation as:
h = (3/4) * h_previous
where h_previous is the height of the previous bounce.
We know that the initial height of the ball is 768 inches, and we want to find the number of bounces when the height of the bounce is less than 9 inches. We can set up an inequality to represent this situation:
h < 9
Substituting the expression for h from the equation above, we get:
(3/4) * h_previous < 9
Now, we can start with the initial height of 768 inches and keep applying the equation for each bounce until the height of the bounce is less than 9 inches.
1st bounce:
h = (3/4) * 768 = 576 inches
2nd bounce:
h = (3/4) * 576 = 432 inches
3rd bounce:
h = (3/4) * 432 = 324 inches
4th bounce:
h = (3/4) * 324 = 243 inches
5th bounce:
h = (3/4) * 243 = 182.25 inches
6th bounce:
h = (3/4) * 182.25 = 136.6875 inches
7th bounce:
h = (3/4) * 136.6875 = 102.515625 inches
8th bounce:
h = (3/4) * 102.515625 = 76.88671875 inches
9th bounce:
h = (3/4) * 76.88671875 = 57.6650390625 inches
10th bounce:
h = (3/4) * 57.6650390625 = 43.248779296875 inches
So, the ball will bounce up to a height less than 9 inches after 10 bounces.