College Algebra & Trigonometry, 2018a

280 CHAPTER 5. CONIC SECTIONS - CIRCLE AND PARABOLA
5.3 Applications of the Parabola
A parabola that is rotated around its axis of symmetry to create a three dimensional
object is called a paraboloid. One of the special properties of a parabola is
that any light (or other electromagnetic wave) striking the interior of the parabola
is reflected to the focus. The proof of this is somewhat complex, but this property
makes the paraboloid a very useful shape in application.
The paraboloid is used to make satellite dishes so that the signal from the satellite
is reflected to the center of the dish. This strengthens the signal. Parabolic microphones
are often used at sporting events so that noises on the field can be heard
more clearly on the sidelines. Flashlights and headlights also use this property
in reverse. With the light at the focus of the paraboloid, all the light is reflected
straight ahead, thus concentrating the beam of light.
One last application which has become used more frequently over the last five or
ten years is the use of the paraboloid in solar power. One use of these properties
sets up a tower at the focus of the paraboloid with mirrors banked around the
tower in the paraboloid shape. This focuses all of the sun’s power on the tower
in the center. Often, salt is used in the tower since it has very high melting point.
The salt is melted by the reflected sunlight and flows into a steam turbine. This
has been found to be more efficient than just using standard solar panels.
Example
If a satellite dish is 8 feet across and 3 ft. deep, how far from the bottom of the
dish should the receiver be placed so that it is at the focus of the paraboloid?
First let’s consider what this looks like in two dimensions. Because we are setting
up the graph of this parabola, we can choose to place the vertex at the origin.
This makes things a little easier.
Since the dish was a total of 8 feet across, we split this between the two sides of
the graph, creating the points (4, 3) and (−4, 3). Since the vertex for this parabola
is at the origin, the standard equation is somewhat simplified.