A mirror is in the shape of a paraboloid. If a light source is 6 feet from the base along the axis of symmetry and the opening is 12 feet across, how deep should the mirror be?
To determine the depth of the mirror, we first need to understand the shape of a paraboloid and how it relates to the given information.
A paraboloid is a three-dimensional shape that is formed by rotating a parabola around its axis of symmetry. It has a characteristic opening (or diameter) and a certain depth (or focal length).
In this case, the opening of the paraboloid is given as 12 feet across, and the distance from the light source to the base along the axis of symmetry is given as 6 feet.
To find the depth of the mirror, we can make use of the mirror's focal length property. The focal length of a parabolic mirror is equal to one-fourth of the diameter or opening.
Therefore, the focal length (F) of the mirror is:
F = 1/4 * opening
F = 1/4 * 12
F = 3 feet
Since the light source is located 6 feet away from the base along the axis of symmetry, the position of the focal point is also 6 feet away from the base.
Now, we can calculate the depth of the mirror by subtracting the focal length from the distance of the light source to the base:
Depth = distance - focal length
Depth = 6 - 3
Depth = 3 feet
Therefore, the mirror should be 3 feet deep to form a paraboloid shape with the given properties.