Question

A searchlight is shaped like a paraboloid of revolution. A light source is located 3 inches from the base along the axis of symmetry. If the opening of the searchlight is 4 feet across, find the depth.

I got 1/3 but was incorrect...

Answer 1

Depth = 4ft

Explanation:

Since located 3 inches from the base, thus focus is (0,3) and vertex at (0,0)

We are told that the opening of the searchlight is 4 feet across.

Thus converting to inches, we have;

4ft = 4 x 12 inches = 48 inches.

Since 48 inches across, it's 24 inches on either side of the vertical axis.

I've attached a drawn diagram of the parabola opening upwards with vertex (0,0), focus (0,3)

Standard form of parabola is;.

x² = 4ay

a is distance from focus to vertex = 3 inches.

Thus,

x² = 4(3)y

x² = 12y

So, y = x²/12

Since the opening is 48 inches across, it needs to cross the parabola at (-24,y) and (24,y)

Thus, the depth of the light will be;

y = 24²/12

y = 48 inches or 4 ft