Answer
Owner should charge $\$15$ to maximize his money to $\$900$.
Work Step by Step
In order to increase the price by $0.25$, the price of the ticket will be $10+0.25x$ and the price of attendance would decrease by $1$ (that is $80-x$).
Therefore, the money made by the owner is,
$\begin{align}
& \text{Money = attendance }\times \text{ price} \\
& \text{=}\left( 80-x \right)\left( 10+0.25x \right) \\
& =800+20x-10x-0.25{{x}^{2}} \\
& =-0.25{{x}^{2}}+10x+800
\end{align}$
Compare it with the standard equation of the parabola $y=a{{x}^{2}}+bx+c$.
$a=-0.25,b=10,c=800$
Now, the vertex of the parabola is,
$\begin{align}
& x=\frac{-b}{2a} \\
& =\frac{-10}{2\left( -0.25 \right)} \\
& =\frac{-10}{-0.5} \\
& =20
\end{align}$
So, when $x=20$, the money made by the owner is a maximum:
Therefore, the money made by the owner is:
$\begin{align}
& -0.25{{x}^{2}}+10x+800=-0.25{{\left( 20 \right)}^{2}}+10\left( 20 \right)+800 \\
& =-100+200+800 \\
& =900
\end{align}$
And the price would be:
$\begin{align}
& \text{Price}=10+0.25\left( 20 \right) \\
& =10+5 \\
& =\$15\end{align}$
Therefore, the owner should charge $\$15$ to maximize his money to $\$900$.
