a certain ball is dropped from a height of x feet. Suppose the ball is dropped from 10 feet and is caught exactly when it touches the ground

We can't give a definitive distance, because it depends upon the rebound height  of the ball. We can model the situation as follows:

The ball travels 10 ft and then strikes the ground. It then rebounds to a height of 10r, where r is  some value between 0 and 1. And it falls this distance again before striking the ground.

So, the ball has traveled 10 + 2*10r feet.

And on the next bounce it rises to a height of (10r)r = 10r² feet ..and it also falls this far before striking the ground again.

So, after two bounces, the ball has traveled  10 + 2*10r + 2*10r²  feet = 10 + 20 (r + r²) feet

So, the total distance traveled after 30 bounces is just  10 + 20(r + r² + r³ +......+ r²⁹ + r³⁰) feet

And we can rewrite this as:     -10 + 20 (1 + r + r² + r³ +......+ r²⁹ + r³⁰ )  ...  and this can be simplified to.......

Total Distance  =  -10 + [ 20 / (1 - r) ]  where  0 < r < 1