Answer
$9$ seconds.
Work Step by Step
The function modelling the rocket's height above the water (in feet) is
$s(t)=-16t^2+128t+144$
where $t$ is the time in seconds.
At $0$ second the height of the rocket is the height of the cliff:
$s(t)=144\; feet$.
$s(0)=-16(0)^2+128(0)+144$
$s(0)=144\; feet$
When the rocket hits the water the value of $s(t)$ is zero.
The function will be
$0=-16t^2+128t+144$
Divide both sides by $16$.
$\frac{0}{16}=\frac{-16t^2}{16}+\frac{128t}{16}+\frac{144}{16}$
Simplify.
$0=-t^2+8t+9$
Rewrite the middle term $8t$ as $9t-t$
$0=-t^2+9t-t+9$
Group terms.
$0=(-t^2+9t)+(-t+9)$
Factor from each term.
$0=-t(t-9)-1(t-9)$
Factor out $(t-9)$.
$0=(t-9)(-t-1)$
Set both factors equal to zero.
$t-9=0$ or $-t-1=0$
Isolate $t$.
$t=9$ or $t=-1$
Take the positive value as $t\geq 0$.
The rocket will take $9$ seconds to hit the water.
