Answer
$s(t)=-16t^{2}+16t+185$
Zenith is reached after $0.5$ seconds,
at height of $189$ ft (4 ft above the tower).
Work Step by Step
The function of velocity is the derivative of $s(t),$ the function of position.
$v(t)=s'(t)=-32t+16$
$s(t)=\displaystyle \int(-32t+16)dt=-32\cdot\frac{t^{2}}{2}+16t+D$
$=-16t^{2}+16t+D$
Given that $ s(0)=+185\qquad$(above ground is a positive position)
$s(t)=-16t^{2}+16t+D$
$185=0+0+D$
$D=185$
Thus,
$s(t)=-16t^{2}+16t+185$
At the zenith, velocity becomes momentarily zero.
This happens when
$-32t+16=0$
$16=32t$
$0.5=t$
Now, we find the position when $t=0.5 \left[\begin{array}{ll}
s(0.5) & =-16(0.5)^{2}+16(0.5)+185\\
& =-4+8+185\\
& =189
\end{array}\right]$
Zenith is reached after $0.5$ seconds,
at height of $189$ ft (4 ft above the tower).
