A parabolic flashlight reflector is to be 12 inches across and 4 inches deep. Where should the lightbulb be placed?

1 Answer
Douglas K.
Nov 7, 2017

The bulb should be placed in the center of the reflector at a point 2.25 inches up from the bottom of the reflector.

Explanation:

The cross-section of the parabolic reflector forms a parabola.
Let the vertex of the parabola be the point #(0,0)#, therefore, the equation of the parabola is:

#y = 1/(4f)x^2#

where #f# is vertical distance from the vertex to the focal point.

The cross-section of the 12 in diameter circle at the top of the reflector tells us that the x values are #+-6#, when #y = 4#. Substitute the point #(6,4)# into the equation:

#4 = 1/(4f)6^2 #

#f = 6^2/4^2#

#f = 2.25#

The bulb should be placed in the center of the reflector at a point 2.25 inches up from the bottom of the reflector.

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