A light house uses a parabolic reflector that is 1 m in diameter how deep should the reflectpr be if the light source is placed halfwayy between the vertex and the plane of the rim?

Question

mathmale · Answer

If the (general) shape of the reflector is that of a parabola, and if we assume that the parabolic reflector opens UP, then the general form of the equation of that parabola is

$$4py=x ^{2}$$

mathmale · Answer

where p represents the distance between the vertex (0,0) and the focus (0, (d/2))   (Note:  d is the unknown depth of the parabolic reflector.)

We need to find d, the depth of the parabolic reflector.

Note that if the diameter of the reflector at the rim is 1 meter (which is a given), then the radius of the the rim is 1/2 meter.  This is an x value.  The reflector will be d meters deep, as measured downward from the rim to the x-axis (that is, parallel to the y axis).

Thus, starting from $$4py=x ^{2}$$
with y=d=depth of reflextor, and x=1/2 meter=radius of rim of reflector, and p=d/2,

$$4(\frac{ d }{ 2 })(d)=(\frac{ 1 }{ 2 })^{2}.$$
Just solve this for d=the depth of the reflector.

mathmale · Answer

It may help you to visualize what's happening here if you were to sketch the parabolic reflector on cartesian coordinates and label everything:
d=depth of reflector
d/2=distance of focus from vertex (0,0)
x=1/2 meter = radius of reflector rim
reflector diameter at rim = 1 meter