SOLUTION: A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 32 feet. If the distance across the top of the mirror is 78 inches, how deep is

Algebra ->  Finance -> SOLUTION: A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 32 feet. If the distance across the top of the mirror is 78 inches, how deep is      Log On


   

Question 1155426: A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 32 feet. If the distance across the top of the mirror is 78 inches, how deep is the mirror in the center?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Consider the origin of a coordinate system at the vertex of the parabola. Then the equation of the parabola is

y+=+%281%2F%284p%29%29x%5E2

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

y+=+%281%2F128%29x%5E2

The distance across the top of the mirror is 78 inches, or 13/2 feet; halfway across is 13/4 feet.

The depth of the mirror in the center is the y value at the edge of the top of the mirror:

y+=+%281%2F128%29%2813%2F4%29%5E2 = 0.08252 feet to several decimal places, or 0.990 inches.

ANSWER: the mirror is just under 1 inch deep at its center.