Question 1119014
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One way to set this up to make the calculations as simple as possible is to put the center of the arch at (0,3); then the bases of the arch are at (-3,0) and (3,0).<br>
With the vertex of the parabola at (0,3), the equation of the parabola is of the form<br>
{{{y = ax^2+3}}}<br>
for some constant a.<br>
To find the value of a, use either of the other known point on the parabola:<br>
{{{0 = a(3^2)+3}}}
{{{0 = 9a+3}}}
{{{9a = -3}}}
{{{a = -1/3}}}<br>
So the equation of the parabola is<br>
{{{y = (-1/3)x^2+3}}}<br>
To find the length of the beam, set y=2 and solve for x.  The length of the beam will be the difference between the two x values.<br>
{{{2 = (-1/3)x^2+3}}}
{{{-1 = (-1/3)x^2}}}
{{{x^2 = 3}}}
{{{x = sqrt(3)}}}  or  {{{x = -sqrt(3)}}}<br>
The length of the beam in meters is the difference between sqrt(3) and -sqrt(3), which is 2*sqrt(3), or 3.464 to 3 decimal places.