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

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 $\left(0 , 0\right)$, therefore, the equation of the parabola is:

$y = \frac{1}{4 f} {x}^{2}$

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 $\pm 6$, when $y = 4$. Substitute the point $\left(6 , 4\right)$ into the equation:

$4 = \frac{1}{4 f} {6}^{2}$

$f = {6}^{2} / {4}^{2}$

$f = 2.25$

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