A ball is dropped from a height of 16 meters and bounces back to 3/4 of its previous height each time
The correct answer and explanation is:
This is an example of a geometric series problem. A ball is dropped from a height of 16 meters and bounces back to 3/4 of its previous height each time.
To find the total distance the ball travels, we can break it down into two parts:
- Initial fall: The ball falls 16 meters initially.
- Subsequent bounces: After hitting the ground, the ball bounces back to 3/4 of its previous height. Therefore, the ball goes up to 12 meters, then falls back down the same distance, and continues this pattern.
The total distance traveled is the sum of the following:
- The first fall: 16 meters
- The first bounce up and down: 2 * (3/4 * 16) = 24 meters
- The second bounce up and down: 2 * (3/4 * 3/4 * 16) = 18 meters
- The third bounce up and down: 2 * (3/4 * 3/4 * 3/4 * 16) = 13.5 meters
The series continues indefinitely, with each term being 3/4 of the previous one.
To find the total distance traveled, we sum the infinite geometric series for the bounces. The sum of an infinite geometric series is given by the formula: S=a1−rS = \frac{a}{1 – r}
where a is the first term and r is the common ratio.
In this case:
- a is 16 (the first drop).
- r is 3/4 (the bounce ratio).
The total distance traveled for the bounces is: S=161−3/4=161/4=64S = \frac{16}{1 – 3/4} = \frac{16}{1/4} = 64
Thus, the total distance traveled is: 16+64=80 meters16 + 64 = 80 \text{ meters}
So, the total distance the ball travels is 80 meters.