First, let's calculate the rocket velocity when it stops accelerating, using Torricelli's equation:
[tex]\begin{gathered} V^2=V^2_0+2\cdot a\cdot d \\ V^2=25.6^2+2\cdot2.34\cdot231 \\ V^2=655.36+1081.08 \\ V^2=1736.44 \\ V=41.67\text{ m/s}^2 \end{gathered}[/tex]After the engine stops, the total acceleration will be the gravity acceleration, with a downwards direction and magnitude of 9.81 m/s².
The maximum height occurs when the velocity is zero, so we can use the formula below:
[tex]\begin{gathered} V=V_0+a\cdot t \\ 0=41.67-9.81\cdot t \\ 9.81t=41.67 \\ t=\frac{41.67}{9.81} \\ t=4.25\text{ s} \end{gathered}[/tex]Now we need to calculate the time spent accelerating (engine turned on):
[tex]\begin{gathered} V=V_0+a\cdot t \\ 41.67=25.6+2.34\cdot t \\ 2.34t=41.67-25.6 \\ 2.34t=16.07 \\ t=\frac{16.07}{2.34} \\ t=6.87\text{ s} \end{gathered}[/tex]Therefore the total time, since lift-off until maximum height, is:
[tex]\begin{gathered} t=6.87+4.25 \\ t=11.12\text{ s} \end{gathered}[/tex]The rocket reaches the maximum height 11.12 seconds after lift-off.