Question

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

Answer 1

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.