You may have made your mistake because you didn't take into account that the
ball drops one more time than it rises.  The second solution above is correct,
but you may not have studied the infinite geometric series, since your problem
stops at the 10th bounce.

If D represents a downward drop of the ball and U represents a upward bounce
of the ball, then the ball's path goes DUDUDUDUDUDUDUDUDUD.  Break it up as: 

D UD UD UD UD UD UD UD UD UD

The initial D is 15 meters. So let's keep that separate and add it in after
we've calculated the other distances the ball travels, which is

   UD UD UD UD UD UD UD UD UD

After the first "D", the ball does only 9 more "UD"'s.

The very first time the ball goes upward, it goes up 60% of 15 meters or 9
meters.
Then it drops 9 meters so that means the first total "UD" is 9+9=18 meters. 

So the geometric series has first term a1 = 18 meters and common ratio r = 60%
or r = 0.6

The formula for the sum of a geometric series with n=9, a1=18 and r=0.6 is

}




That works out with a calculator to be 44.54650368, then when we
add the initial drop of 15 meters, we get

59.54650368 which rounds to 59.5 meters.

Edwin