Answer
a) $3.5$ feet
b) $0.7927$ seconds and $3.135$ seconds
c) $69.52$ feet
d) $4.12$ seconds
Work Step by Step
Given \begin{equation}
h(t)=-16 t^2+65 t+3.5.
\end{equation} a) Set $t=0$ to find the height of the ball when it is hit.
\begin{equation}
\begin{aligned}
h(0) & =-16 \cdot(0)^2+65 \cdot(0)+3.5 \\
& =3.5.
\end{aligned}
\end{equation} The ball is hit at a height of $3.5$ ft .
b) Set $h(t)=50$ to find the time.
\begin{equation}
\begin{aligned}
&-16 t^2+65 t+3.5=50 \\
& -16 t^2+65 t+3.5-50=0 \\
& \frac{-16 t^2+65 t-46.5}{-16}=\frac{0}{-16} \\
& t^2-4.0625 t+2.90625=0.
\end{aligned}
\end{equation} Use the quadratic formula with $a= 1,\quad b= -4.0625\ ,\quad c = 2.90625$.
\begin{equation}
\begin{aligned}
&t= \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\
& t=\frac{-(-4.0625) \pm \sqrt{(-4.0625)^2-4(1)(2.90625)}}{2(1)} \\
& =\frac{4.0625 \pm \sqrt{4.87890625}}{2}\\
&= \frac{4.0625\pm 2.2088}{2}\\
&= 2.031\pm 1.1044\\
\end{aligned}
\end{equation} \begin{equation}
\begin{aligned}
& t=2.031-1.1044=0.7927 \\
& t=2.031+1.1044=3.135
\end{aligned}
\end{equation} The ball reached a height of 50 feet after $0.7927$ seconds and after $3.135$ seconds.
c) The time when the ball reached its maximum height is found from:
\begin{equation}
t=\frac{-b}{2 a}=\frac{-65}{2(-16)}=2.031
\end{equation} The ball reached its maximum height aster $1.75$ seconds.
The maximum height of the ball is found from:
\begin{equation}
\begin{aligned}
h(2.031) & =-16 \cdot(2.031)^2+65 \cdot(2.031)+3.5 =69.52.
\end{aligned}
\end{equation} The maximum height of the ball is $69.52$ feet.
d) Set $h=0$ to find the time the ball hit the ground.
\begin{equation}
\begin{aligned}
&-16 t^2+65 t+3.5&= 0.
\end{aligned}
\end{equation} \begin{equation}
\begin{aligned}
& t=\frac{-(65) \pm \sqrt{(65)^2-4(-16)(3.5)}}{2(-16)} \\
& =\frac{-65 \pm \sqrt{4449}}{-32}\\
&=-(-2.03125 \pm 2.08440 ).
\end{aligned}
\end{equation} \begin{equation}
\begin{aligned}
t & = -(-2.03125 - 2.08440 ) \\
& =4.12 \\
t & =-(-2.03125 + 2.08440 ) \\
& =-0.053.
\end{aligned}
\end{equation} The ball will hit the ground after about $4.12$ seconds later after being hit.
