Given:
The height equation is,
[tex]h(t)=-16t^2+144t+6[/tex]
Explanation:
For maximum/minimum of a function, the first derivative of function is 0.
Differentiate the function with respect to x.
[tex]\begin{gathered} \frac{d}{dt}h(t)=\frac{d}{dt}(-16t^2+144t+6) \\ =-32t+144 \end{gathered}[/tex]
For maximum and minimum,
[tex]\begin{gathered} -32t+144=0 \\ t=\frac{144}{32} \\ =4.5 \end{gathered}[/tex]
So rocket reach it maximum height after 4.5 seconds of launch.
Substitute 4.5 for t in the equation to determine the maximum reached by rocket.
[tex]\begin{gathered} h(4.5)=-16(4.5)^2+144\cdot4.5+6 \\ =-324+648+6 \\ =330 \end{gathered}[/tex]
So maximum height of rocket is 330 feet.
