In this problem, we want to apply the use of a quadratic function.
We are given the following equation to model the path of a rocket:
[tex]-16t^2+144t=0[/tex]
To find how long it takes the rocket to hit the ground, the equation must be equal to 0. Luckily, that's already true.
Begin by identifying the greatest common factor of
[tex]-16t^2\text{ and }144t.[/tex]
We see that both terms have 16 and t in common:
[tex]GCF(-16t^2,144t)=16t[/tex]
We can factor that expression out from each term to get:
[tex]-16t^2+144t=16t(-t+9)[/tex]
Now we have the equation:
[tex]16t(-t+9)=0[/tex]
Applying the zero product property, we can split this into two equations:
[tex]16t=0\text{ and }-t+9=0[/tex]
Solve the first equation by dividing by 16 on both sides:
[tex]\begin{gathered} 16t=0 \\ \\ t=0 \end{gathered}[/tex]
Solve the second equation by adding t to both sides:
[tex]\begin{gathered} -t+9=0 \\ \\ 9=t \end{gathered}[/tex]
Since we want to know how long it takes the rocket to hit the ground after takeoff, we can ignore the first solution, which represents 0 seconds.
It took the rocket 9 seconds to hit the ground.
