Question

A ball is dropped from a height of 60 meters and always rebounds one-third the distance from which it was dropped. If you let the ball bounce forever, what is the total distance it would travel until it comes to rest? A) 80 meters B) 90 meters C) 100 meters D) 120 meters

Answer 1

The ball would travel a total distance of 90 meters until it comes to rest.

Step-by-step explanation:

To find the total distance the ball would travel until it comes to rest, we can use the concept of an infinite geometric series. Each bounce of the ball can be represented as a fraction of the previous bounce. Since the ball always rebounds one-third the distance from which it was dropped, the distance traveled after each bounce can be calculated as follows:

1. Bounce 1: 60 meters
2. Bounce 2: 60 * (1/3) = 20 meters
3. Bounce 3: 20 * (1/3) = 6.67 meters

To find the sum of this infinite geometric series, we can use the formula for the sum of an infinite geometric series, which is:

S = a / (1 - r)

where:

S = sum of the series

a = first term of the series

r = common ratio

In this case, the first term (a) is 60 meters and the common ratio (r) is 1/3. Plugging these values into the formula, we get:

S = 60 / (1 - 1/3) = 60 / (2/3) = 90 meters

Therefore, the total distance the ball would travel until it comes to rest is 90 meters. So the correct answer is B) 90 meters.

Answer 2

The total distance the ball would travel until it comes to rest is 120 meters (option D).

How to determine the total distance traveled by the ball

To determine the total distance traveled by the ball until it comes to rest, sum up the distances covered during each bounce.

Given:

Initial height = 60 meters

The ball rebounds one-third the distance from which it was dropped.

Let's calculate the distance traveled during each bounce:

1st bounce: The ball falls from 60 meters and rebounds one-third of 60 meters, which is (1/3) * 60 = 20 meters.

So, the total distance covered during the 1st bounce is 60 meters (fall) + 20 meters (rebound) = 80 meters.

2nd bounce: The ball falls from 20 meters (rebound height from the 1st bounce) and rebounds one-third of 20 meters, which is (1/3) * 20 = 6.67 meters (rounded to two decimal places).

So, the total distance covered during the 2nd bounce is 20 meters (fall) + 6.67 meters (rebound) = 26.67 meters.

3rd bounce: The ball falls from 6.67 meters (rebound height from the 2nd bounce) and rebounds one-third of 6.67 meters, which is (1/3) * 6.67 ≈ 2.22 meters (rounded to two decimal places).

So, the total distance covered during the 3rd bounce is 6.67 meters (fall) + 2.22 meters (rebound) ≈ 8.89 meters.

The pattern continues with each bounce, where the fall height decreases by one-third each time.

To find the total distance covered until the ball comes to rest, sum up the distances covered during each bounce:

Total distance = 80 meters (1st bounce) + 26.67 meters (2nd bounce) + 8.89 meters (3rd bounce) + ...

This forms a geometric series with a common ratio of 1/3.

Using the formula for the sum of an infinite geometric series, calculate the total distance:

Total distance = a / (1 - r)

where a is the first term (80 meters) and r is the common ratio (1/3).

Total distance = 80 / (1 - 1/3)

Total distance = 80 / (2/3)

Total distance = 80 * (3/2)

Total distance = 120 meters

Therefore, the total distance the ball would travel until it comes to rest is 120 meters (option D).