Problem 63 - College Algebra with Corequisite Support

Question

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?

Step-by-Step Solution

Answer

The receiver should be placed 2.25 feet from the vertex of the dish.

Step 1: Understanding the Problem

We need to find the location of the focus of a parabolic dish that is 12 feet wide and 4 feet deep. The receiver must be placed at the focus of this parabola, which is a feature of its geometric property.

Step 2: Writing the Parabolic Equation

The general equation for a parabola with its vertex at the origin and opening upwards is given by \[ y = \frac{1}{4p}x^2 \]where \(p\) is the distance from the vertex to the focus.

Step 3: Substitute the Known Values

We substitute the point (6, 4) into the parabolic equation. The x-coordinate is half of the dish's width (6 feet), and the y-coordinate is the depth (4 feet). \[ 4 = \frac{1}{4p}(6)^2 \]

Step 4: Solve for \(p\)

From the equation \[ 4 = \frac{36}{4p} \],we solve for \(p\).This simplifies to \[ 4 = \frac{36}{4p} \],multiplying both sides by \(4p\) gives \[ 16p = 36 \],and thus \[ p = \frac{36}{16} = 2.25 \].

Step 5: Interpret the Result

The focus, where the receiver should be placed, is 2.25 feet from the vertex of the parabola along the axis of symmetry, towards the opening of the dish.

Key Concepts

Paraboloid of Revolution

A paraboloid of revolution is created by rotating a parabola around its axis of symmetry. Imagine taking a spoon and spinning it around its handle. This creates a three-dimensional shape that resembles a satellite dish. These shapes are used in real life to collect signals, like those from a satellite.

Paraboloids of revolution have the fascinating ability to focus parallel rays of light or signals to a single point, if aimed directly at the dish. This property makes them extremely useful for applications in both telescopes and communication devices. The position where all incoming signals converge is called the focus.

Focus of a Parabola

The focus of a parabola is a very special point. It lies along the axis of symmetry of the parabola. Mathematically, it is located at a distance \(p\) from the vertex of the parabola, where \(p\) is the focal length.

In the context of a satellite dish, this focus is where the receiver collects signals. Since the parabola has the property of reflecting all parallel beams through this point, the signals from a satellite can be gathered efficiently at this focus. For the given exercise, the receiver must be placed precisely 2.25 feet from the vertex of the dish along its axis of symmetry.

Parabolic Dish

A parabolic dish refers to structures that have a dish-shaped surface in the form of a paraboloid of revolution. These dishes are prominently used in satellite communications and radio telescopes due to their unique shape.

They provide the advantage of focusing all incoming electromagnetic waves onto the receiver located at the focus. This high degree of focus enhances the quality and strength of the signal being received. The size of the dish is often proportional to the ability to capture weaker signals over greater distances. In the exercise, the dish is 12 feet wide and 4 feet deep, with its specifications guiding you to find where the receiver, or focus, should be set.

Vertex of a Parabola

The vertex of a parabola is the point where the parabola makes its sharpest turn. It serves as the point of origin in a parabola's equation when centered at the origin. In the equation \(y = \frac{1}{4p}x^2\), the vertex is at the origin, making calculations straightforward.

For the parabolic dish exercise, the vertex is the point at the center-bottom of the dish, which is the point from which the vertical depth of the dish is measured. It's crucial as it provides a reference for determining the position of the focus, ultimately leading to the ideal placement of the receiver. The accuracy in this positioning affects how well signals are captured by the dish.