Answer: The ball is dropped from a height of 10 ft, so it first travels down 10 ft until it hits the ground. The distance traveled in this first part is 10 ft.
On the first bounce, the ball rebounds to one-fifth of its previous height, which is 2 ft (since 10/5 = 2). The ball then travels up 2 ft and back down 2 ft to the ground, for a total distance traveled of 10 + 2 + 2 = 14 ft.
On the second bounce, the ball rebounds to one-fifth of its previous height, which is 2/5 ft (since 2/5 x 2 = 4/5). The ball then travels up 4/5 ft and back down 4/5 ft to the ground, for a total distance traveled of 2/5 + 4/5 + 4/5 = 2 ft.
On the third bounce, the ball rebounds to one-fifth of its previous height, which is 4/25 ft (since 4/25 x 2/5 = 8/125). The ball then travels up 8/125 ft and back down 8/125 ft to the ground, for a total distance traveled of 4/25 + 8/125 + 8/125 = 0.32 ft (rounded to two decimal places).
The ball will continue to bounce, getting closer and closer to the ground with each bounce. We can calculate the total distance traveled by summing the distances traveled on each bounce:
10 + 14 + 2 + 0.32 + ...
To calculate the sum of this infinite series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
In this case, a = 10 (the distance traveled on the first drop), and r = 1/5 (the fraction by which the height decreases on each bounce). Plugging in these values, we get:
S = 10 / (1 - 1/5)
= 12.5
So the total distance traveled by the ball is 12.5 ft.
Explanation: