A bouncy ball is dropped such that the height of its first bounce is 6.5 feet and each successive bounce is 78% of the previous bounce's height. What would be the height of the 10th bounce of the ball? Round to the nearest tenth (if necessary).


The height of each bounce follows a geometric sequence because each bounce is a constant percentage of the previous one. The formula for the \(n\)-th term of a geometric sequence is:

\[
h_n = h_1 \cdot r^{(n-1)}
\]

Where:
- \(h_n\) is the height of the \(n\)-th bounce,
- \(h_1 = 6.5\) feet (the first bounce),
- \(r = 0.78\) (the ratio of successive bounces),
- \(n = 10\) (for the 10th bounce).

Substituting the values:

\[
h_{10} = 6.5 \cdot 0.78^{9}
\]

Let me calculate that for you.

The height of the 10th bounce is approximately 0.7 feet.

Would you like more details or have any questions about this?

Here are 5 related questions:
1. How many bounces would it take for the ball's height to drop below 1 foot?
2. What is the total vertical distance traveled by the ball after 10 bounces?
3. What percentage of the original height would the 15th bounce be?
4. If the ball were dropped from a different height, how would that affect the bounce sequence?
5. How does the ratio of 78% affect the speed at which the ball's height decreases?

**Tip:** In geometric sequences, the common ratio (\(r\)) directly influences how fast the terms decay or grow.

A bouncy ball is dropped such that the height of its first bounce is 6.5 feet and each successive bounce is 78% of the previous bounce's height. What would be the height of the 10th bounce of the ball? Round to the nearest tenth (if necessary).

Learn how to calculate the height of the 10th bounce of a ball dropped from 6.5 feet, with each bounce reaching 78% of the previous height. This problem applies geometric sequences and exponential decay, making it perfect for students in Grades 9-10.

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