[tex]\begin{gathered} 3m\text{ - }\frac{2}{3}(3m)\text{ - \lparen}\frac{2}{3})^x(3m)=\text{ 0} \\ 3m\text{ - }2m\text{ }^{\text{ }}\text{- \lparen}\frac{2}{3})^x(3m)\text{ = 0} \\ 1m=\text{\lparen}\frac{2}{3})^x(3m) \\ \frac{1}{3}=(\frac{2}{3})^x \\ log\frac{1}{3}=x\text{ }log(\frac{2}{3}) \\ \frac{log\frac{1}{3}}{log\frac{2}{3}}=x \\ x=\text{ 2.70} \end{gathered}[/tex]
So it will bounce2/3 to the 2.71 times.
So the distance is:
[tex]\begin{gathered} 3m+2m+(\frac{2}{3})^{2.71}(3m) \\ =5m\text{ + \lparen0.33\rparen\lparen3m\rparen} \\ =5m+0.99m \\ =5.99m \end{gathered}[/tex]
