To find the total distance traveled by the ball if it bounces infinitely many times, we can use the concept of an infinite geometric series.
In this case, the distance traveled by the ball during each downward and upward motion forms a geometric sequence with a common ratio of 2/3. The first term is the initial height of 20 feet, and each subsequent term is 2/3 times the previous term.
The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
Where: S is the sum of the series, a is the first term, r is the common ratio.
In this case, a = 20 feet and r = 2/3.
Plugging in the values, we have:
S = 20 / (1 - 2/3)
Simplifying:
S = 20 / (1/3)
S = 60 feet
Therefore, the total distance traveled by the ball, if it bounces infinitely many times, is 60 feet.