Question 1155426
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Consider the origin of a coordinate system at the vertex of the parabola.  Then the equation of the parabola is<br>
{{{y = (1/(4p))x^2}}}<br>
where p is the distance from the vertex to the focus.  Given that that distance is 32 feet, the equation for this parabolic mirror is<br>
{{{y = (1/128)x^2}}}<br>
The distance across the top of the mirror is 78 inches, or 13/2 feet; halfway across is 13/4 feet.<br>
The depth of the mirror in the center is the y value at the edge of the top of the mirror:<br>
{{{y = (1/128)(13/4)^2}}} = 0.08252 feet to several decimal places, or 0.990 inches.<br>
ANSWER: the mirror is just under 1 inch deep at its center.<br>