A satellite dish has the shape of a paraboloid. The signals...

Name: Problem 39, Chapter 8: College Algebra and Trigonometry
Uploaded: 2021-10-25T19:15:17-08:00
Duration: 1 min 27 s
Channel: Sheryl Ezze
Description: A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1 foot deep at its vertex, determine the location (distance from the vertex of the dish) of its focus.

A satellite dish has the shape of a paraboloid. The signals that it receives are reflected to a receiver that is located at the focus of the paraboloid. If the dish is 8 feet across at its opening and 1 foot deep at its vertex, determine the location (distance from the vertex of the dish) of its focus.

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Key Concepts

Parabola

A parabola is a type of conic section defined as the collection of all points in the plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). This geometric property gives parabolas their unique reflective characteristics, which are exploited in satellite dishes and other applications.

Focus of a Parabola

The focus is a key point associated with a parabola, where rays parallel to the axis of symmetry reflect and converge. In the standard form of a parabola, given by y = ax², the focus lies at the point (0, 1/(4a)). This relationship allows one to determine the focus’s location once the coefficient 'a' is known.

Standard Form of a Parabola

The standard form of a parabola, typically expressed as y = ax² when the vertex is at the origin, is crucial for linking the algebraic representation to its geometric properties. This formulation makes it straightforward to calculate important features such as the focus and the width or depth of the parabola.

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