Answer
$s(t)=-16t^{2}+24t+185$
Zenith is reached after $0.75$ seconds,
at height of $194$ ft ($9$ ft above the tower).
Work Step by Step
The function of velocity is the derivative of $s(t),$ the function of position.
$v(t)=s'(t)= -32t+24$
$s(t)=\displaystyle \int(-32t+24)dt=-32\cdot\frac{t^{2}}{2}+24t+D$
$=-16t^{2}+24t+D$
Given that $ s(0)=+185\qquad$(above ground is a positive position),
$185=0+0+D$
$D=185$
Thus,
$s(t)=-16t^{2}+24t+185$
At the zenith, velocity becomes momentarily zero.
This happens when
$-32t+24=0$
$t=24/32=0.75$ s
Now, we find the position when $t=0.75$
$s(0.75)=-16(0.75)^{2}+24(0.75)+185=194\ \ ft$
