Question 1196473
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The vertex is to be at the top of the arch; let the origin of a coordinate system be at the center of the base of the arch.<br>
With a height of 24 at the center of the arch, the coordinates of the vertex are (0,24).<br>
And with a width of 32 at the base, the coordinates of the endpoints of the arch are (-16,0) and (16,0).<br>
With those conditions, the equation of the parabolic arch is {{{y=ax^2+24}}}, where a is a (negative) constant to be determined.<br>
That constant can be determined using either endpoint of the arch.<br>
{{{0=a(16^2)+24}}}
{{{0=256a+24}}}
{{{a=-24/256=-3/32}}}<br>
The equation of the arch is {{{y=(-3/32)x^2+24}}}<br>
The object that is to pass through the arch has a width of 25, so it will extend 12.5 units each side of the center of the arch.  To find the maximum height of the object, evaluate the equation for x=12.5.  The answer is an ugly number, so I'll let you do that part.  You might want to use a graphing calculator instead of working with the ugly arithmetic....<br>