Answer
a) The maximum height attained by the shot is $35\text{ feet}$, and the distance at which it occurs is $18.35\text{ feet}$.
b) The maximum horizontal distance is $77.8\text{ feet}$.
c) The shot was released from $\underline{\text{6}\text{.1 feet}}$.
Work Step by Step
(a)
We have to compare the equation $f\left( x \right)=-0.01{{x}^{2}}+0.7x+6.1$with the standard equation:
$f\left( x \right)=a{{x}^{2}}+bx+c$ ,
It gives:
$a=-0.01,\text{ }b=0.7,\text{and }c=6.1.$
As $a<0,$ the maximum height will occur at $x=\frac{-b}{2a}$.
$\begin{align}
& x=\frac{-0.7}{2\left( -0.01 \right)} \\
& =\frac{-0.7}{-0.02} \\
& =35
\end{align}$
So, the maximum height of the shot occurs at 35 feet from the point of release.
The distance travelled by the shot is:
$\begin{align}
& f\left( 35 \right)=-0.01{{\left( 35 \right)}^{2}}+0.7\left( 35 \right)+6.1 \\
& =18.35
\end{align}$
Thus, the maximum height of the shot occurs at $35\text{ ft}$ from the point of release and the distance is $18.35\text{ ft}$.
(b)
The maximum horizontal displacement is determined by calculating the x-intercept.
To find the x-intercept, replace $f\left( x \right)\text{ by 0}$ and solve the quadratic equation obtained.
$0=-0.01{{x}^{2}}+0.7x+6.1$
And solve the equation using quadratic formula:
$\begin{align}
& x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-0.7\pm \sqrt{{{\left( 0.7 \right)}^{2}}-4\left( -0.01 \right)\left( 6.1 \right)}}{2\left( -0.01 \right)} \\
& =\frac{-0.7\pm \sqrt{0.734}}{-0.02} \\
& =\frac{-0.7\pm 0.8}{-0.02}
\end{align}$
Get, $x=77.8\text{ or }x=-7.8$
As $x\ge 0,$ the maximum horizontal distance of the shot will be $x=77.8\text{ feet}$.
Hence, the maximum horizontal distance is $77.8\text{ feet}$.
(c)
The height from which the shot is released is given by its y-intercept, that is $f\left( 0 \right)$
$\begin{align}
& f\left( 0 \right)=-0.01{{\left( 0 \right)}^{2}}+0.7\left( 0 \right)+6.1 \\
& =6.1
\end{align}$
So, the height of the shot will be $6.1\text{ feet}$.
Thus, the height at which the shot was released is $\text{6}\text{.1 feet}$.
