Calculating the Depth of a Parabolic Reflector

What is the depth of the parabolic reflector based on the given geometry?

The depth of the reflector determined by the geometry of a parabola is approximately 1.56 feet.

Parabolic reflectors are commonly used in various applications, such as satellite dishes and car headlights, due to their unique focusing properties. The geometry of a parabolic reflector plays a crucial role in determining its depth, which affects its performance.

Understanding the Geometry of a Parabolic Reflector

The cross-section of a parabolic reflector has a vertical axis of symmetry, with the vertex at (0,0) and the focus located 4 feet above the vertex. The reflector extends 5 feet to either side of the vertex. To calculate the depth of the reflector, we can utilize the equation of a parabola with a vertical axis of symmetry.

With the vertex at (0,0) and the focus at (0,4), the equation of the parabola is y = 1/16x². This equation is derived from the general formula for a parabola's equation with a vertical axis of symmetry, y = 1/4f *x², where 'f' represents the focal length. In this case, the focal length 'f' is 4, leading to the equation y = 1/16x².

Calculating the Depth of the Parabolic Reflector

By substituting the value x = 5 into the equation y = 1/16x², we can determine the depth of the reflector. This calculation results in y = 1/16(5²) = 1.56 feet. Therefore, based on the given geometry, the depth of the parabolic reflector is approximately 1.56 feet.

This calculation highlights the importance of understanding the geometry of parabolic reflectors in various applications. By applying geometric principles, we can accurately determine key parameters such as depth, which ultimately impacts the performance of the reflector.