A ball is dropped from a height of 3 feet. On each bounce, it returns to a height that is two-thirds of the distance it fell. Find the total vertical distance the ball travels.
A) 9 feet
B) 6 feet
C) 15 feet
D) 38 feet

Question

Quizplus · Accepted Answer

On the first bounce, the ball reaches a height of 2 feet, then falls back to 3 feet. On the second bounce, it reaches a height of 4/3 feet, then falls back to 2 feet. On the third bounce, it reaches a height of 8/9 feet, then falls back to 4/3 feet. This pattern repeats, with the heights achieved on each bounce forming a geometric sequence with a first term of 2 and a common ratio of 2/3.

The total vertical distance the ball travels is equal to the sum of the distances it falls and the distances it rises on each bounce. Since the ball is dropped from a height of 3 feet, it falls a total distance of 3 feet on the first bounce. On the second bounce, it falls a distance of 1 foot (from 2 feet to 1 foot) and rises a distance of 1/3 foot (from 1 foot to 4/3 feet), for a total distance of 4/3 feet. On the third bounce, it falls a distance of 2/3 foot (from 4/3 feet to 2/3 foot) and rises a distance of 4/27 foot (from 2/3 foot to 8/9 feet), for a total distance of 2/27 feet.

The total vertical distance the ball travels is therefore:

3 + 4/3 + 2/27 + ...

This is an infinite geometric series with a first term of 3 and a common ratio of 2/3. The sum of an infinite geometric series with a first term a and a common ratio r (|r| < 1) is a/(1 - r). Therefore:

Total distance = 3/(1 - 2/3) = 9 feet.

A Ball Is Dropped from a Height of 3 Feet