Question

A searchlight is shaped like a parabola. If the light source is located 3 feet from the base along the axis of symmetry and the depth of the searchlight is 4 feet, what should the width of the opening of the searchlight be?

Answer 1

The geometry of a parabolic reflector is such that a light source at the focus is reflected by the parabolic reflector into a parallel beam of light. In this representation the focus is at (0,3) where the coordinate system is a vertical y-axis and a horizontal x-axis (the searchlight is pointing vertically upward).

The general equation of a parabola is 4fy=x² where f is the focal length (focus is at (0,f)).

So the equation is 12y=x², or y=x²/12.

The depth of the searchlight is 4ft, so if we plug in y=4, we get 4=x²/12, x²=48, x=±4√3=±6.9282ft. approx. The width of the opening should be 8√3, about 13.86ft.