check the picture below, that's a parabolic graph of a quadratic equation with a negative leading coeffficient, like this one, the one in the graph is just for initial velocity, but, the highest point is at the vertex for any of these parabolas.
and this one is not exception, so where's the vertex? what's A or the y-coordinate when the vertex is reached?, well, let's check.
[tex]\bf A=24x-x^2\implies A=-1x^2+24x+0\\\\
\textit{ vertex of a vertical parabola, using coefficients}\\\\
\begin{array}{lccclll}
A = &{{ -1}}x^2&{{ +24}}x&{{ +0}}\\
&\uparrow &\uparrow &\uparrow \\
&a&b&c
\end{array}\qquad
\left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)
\\\\\\
\left(-\cfrac{24}{2(-1)}~~,~~0-\cfrac{24^2}{4(-1)} \right)\implies \left(12~~,~~ 0+\cfrac{576}{4} \right)
\\\\\\
\left(\stackrel{width}{12}~~,~~\stackrel{Area}{144} \right)[/tex]
so, that's the vertex, and when the width is that much, the Area reaches a maximum of that much.
