Question

I asked this question yesterday, but I wasn't clear about the response. Could someone please explain it again?

A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of symmetry and the opening is 5 feet across, how deep should the search light be?

I don't see how you can find an equation with only that information.

Answer 1

The focal length is 2 feet. That is the didtanbce from the light-producing arc to the bottom of the mirror. That focal distance equals 1/4a, where a is the constant in the equation y = ax^2 of the equation of the parabola that generates the paraboloidal surface of the searchlight mirror.

The problem should have made clear why a paraboloidal mirror is used and why the light source is placed at the focus. Paraboloids produce the best parallel "collimated" beam of light. Telescope mirrors are paraboloids for the same reason.

Therefore
1/(4a) = 2
That equation comes from basic analytic geometry of parabolas, which you should perhaps review.
4a = 1/2
a = 1/8
y = x^2/8 is the equation of the parabola, if y and x are in feet.
At the top of the parabola, x = 2.5 ft since it is 5 feet across.
Therefore the depth of the parabola is
(2.5)^2/8 = 0.781 feet.