Answer:
53.33 feet
Explanation:
Let's calculate the total distance the ball has traveled after bouncing.
The ball is dropped from a height of 40 feet. During the first fall, it travels the full distance of 40 feet.
Distance fallen = 40 feet
After the first fall, the ball bounces back up but only reaches 1/4 of the distance it fell during the previous fall.
Distance bounced back up = 1/4 * 40 feet = 10 feet
The ball then falls again from the height it bounced back up to (10 feet).
Distance fallen = 10 feet
After the second fall, the ball bounces up again but only reaches 1/4 of the distance it fell during the second fall.
Distance bounced back up = 1/4 * 10 feet = 2.5 feet
The ball then falls again from the height it bounced back up to (2.5 feet).
Distance fallen = 2.5 feet
This process of bouncing and falling repeats indefinitely, but each bounce becomes shorter and shorter as it is only 1/4 of the previous fall.
Now, to find the total distance traveled by the ball, we need to sum up the distances fallen and the distances bounced back up:
Total distance = 40 feet + 10 feet + 2.5 feet + ... (and so on, infinite series)
To calculate the infinite series, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r)
where 'a' is the first term (40 feet) and 'r' is the common ratio (1/4).
Sum = 40 / (1 - 1/4) = 40 / (3/4) = 40 * (4/3) = 160/3 ≈ 53.33 feet
So, the ball has traveled approximately 53.33 feet once it is done bouncing.