Final answer:
The total distance traveled by the ball if it bounces infinitely many times is 52 feet.
Step-by-step explanation:
To find the total distance traveled by the ball, we can consider it as a geometric series. The initial height is 13 feet, and each time the ball bounces, it returns to 3/4 of its previous height. So the distances traveled by the ball after each bounce form a geometric progression with a common ratio of 3/4.
The sum of an infinite geometric series can be calculated using the formula:
Sum = a / (1 - r), where a is the first term and r is the common ratio.
In this case, a = 13 feet and r = 3/4. Plugging these values into the formula, we get:
Sum = 13 / (1 - 3/4) = 13 / (1/4) = 52 feet.
Therefore, the total distance traveled by the ball if it bounces infinitely many times is 52 feet.