Question 1100625
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the parabolic opening of a tunnel is 32 m wide measured from side to side along the ground. 
at the points that are 4 m from each side the tunnel entrance is 6 m high 


determine a equation of the function that models the opening of the tunnel
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<pre>
Put the origin of the coordinate system at the highest point of the arch, the axis "y" directed vertically up.

so the the origin of the coordinate system will be THE VERTEX of the parabola.


You have y = ax^2 as the equation of the parabola with the negative unknown coefficient "a".


The equation for "a" is

y(12) - y(16) = {{{a*12^2}}} - {{{a*16^2}}} = 6,     (1)

saying that  at the distance of 12 = 16-4 meters from the side the height of the tunnel is 6 meters.


Then  a = {{{6/(12^2-16^2)}}} = {{{6/(144-256)}}} = {{{-6/112}}} = {{{-3/56}}}.


Hence and finally, the equation of the parabola is

y = {{{(-3/56)*x^2}}} 

in this coordinate system.   // And, although the problem does not ask about it, the height of the tunnel is  {{{(3/56)*16^2}}} = 13.71 meters.
</pre>

The tutor @Fombitz incorrectly interpreted the condition.