Question

A lighthouse uses a parabolic reflector that is 1 meter in diameter. How deep should the reflector be if the light source is placed halfway between the vertex and plane of the rim?

Answer 1

Using http://en.wikipedia.org/wiki/Parabola#Equations and the attached pic. We have

y= x^2/4p, where p is the focus. I will assume that h and k are zero (see link above for definition of h and k).

We want the light source to be at the focus, p (for maximum reflection), which is to be h/2 meters from the top plane. This means that the focus must be h/2 from the vertex (see diagram).

So p = h/2 or h = 2p.

Therefore, since y= x^2/4p

1 = (r/2)^2 / (2h) = (1/2)^2/(2h)

2h = 1/4

h = 1/8 meters.

The reflector should be 1/8 meters deep.

Answer 2

Sorry, but where did the value for x and y came from? in the part [(2h) = (1/2)^2/(2h)]

Answer 3

There is a small error, because at x=r/2, y = h NOT y=1.

Take the point at the edge of the plane where we know both x and y. That is where x = r/2 and y = h. We also have p = h/2, because we want the light source to be at the focus.

So, using y = x^2/4p
h = (1/2)^2 / 4(h/2) = (1/4) / (2h)

h^2= 1/8

h = (1/8)^(1/2)

h = 1/ (2sqrt(2) ) = sqrt(2)/4

Answer 4

Thank You