Let's determine the height to which a ball will rise after each bounce, given that with each bounce, the ball rises to [tex]\(\frac{2}{3}\)[/tex] of its previous height. The ball is initially dropped from a height of 81 cm.
1. Initial Height:
The ball is dropped from an initial height of 81 cm.
2. Height after the 1st bounce:
After the first bounce, the ball rises to [tex]\(\frac{2}{3}\)[/tex] of the initial height.
[tex]\[
\text{Height after 1st bounce} = 81 \times \frac{2}{3} = 54 \text{ cm}
\][/tex]
3. Height after the 2nd bounce:
After the second bounce, the ball rises to [tex]\(\frac{2}{3}\)[/tex] of the height after the first bounce.
[tex]\[
\text{Height after 2nd bounce} = 54 \times \frac{2}{3} = 36 \text{ cm}
\][/tex]
4. Height after the 3rd bounce:
After the third bounce, the ball rises to [tex]\(\frac{2}{3}\)[/tex] of the height after the second bounce.
[tex]\[
\text{Height after 3rd bounce} = 36 \times \frac{2}{3} = 24 \text{ cm}
\][/tex]
5. Height after the 4th bounce:
Finally, after the fourth bounce, the ball rises to [tex]\(\frac{2}{3}\)[/tex] of the height after the third bounce.
[tex]\[
\text{Height after 4th bounce} = 24 \times \frac{2}{3} = 16 \text{ cm}
\][/tex]
Therefore, the height to which the ball will rise after the fourth bounce is 16 cm.
