Miracle Glass, Inc makes parabolic headlights for many domestic automobiles. If one of its headlights has a parabolic reflector that is 20 in. wide from rim to rim and 12 in. deep, where should its light bulb be placed? Please show work.
Consider the shape on the graph above bounded by the parabolic curve and the horizontal line. This is similar to the shape of the parabolic reflector in question. As you can see, it is 20 units wide from rim to rim and 12 units deep.
The equation of this parabolic curve is `y = ax^2`
To find a, plug in the half-width of the reflector, 20/2 = 10, for x. This has to result in y-value equal to the depth of the reflector, 12:
For x = 10, y = 12. Thus,
`12 = a*10^2`
`a = 12/10^2 = 12/100 = 3/25`
So the equation of the parabolic curve which determines the shape of the reflector is
`y = 3/25 x^2`
The light bulb should be placed in the focus of this parabolic curve. The distance between focal point and the vertex of the parabola (located at the origin, the point (0,0)), equals
`f = 1/(4a)`
`f = 1(4a) = 1/4*1/a = 1/4*25/3 = 25/12`
So the focal point of the parabola is located at the point `(0, 25/12)`
25/12 is about 2.08.
This means the light bulb should be placed approximately 2.08 inches away from the bottom of the reflector.