Problem 58 - Precalculus: Functions and Graphs

Question

A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the receiver be placed to receive the greatest intensity of sound waves?

Step-by-Step Solution

Answer

Place the receiver approximately 2.08 feet from the vertex.

Step 1: Understand the problem

The problem describes a satellite antenna dish which is in the shape of a paraboloid and asks where the receiver should be placed for optimal performance. We need to find the focal point of the paraboloid.

Step 2: Establish the parabola equation

The cross-section of the paraboloid is a parabola. Assume the parabola is positioned with its vertex at the origin (0,0) and opens upward. The dish is 10 feet across the open end, so the point at the edge is (5, 3). The general equation of the parabola can be written as: \(x^2 = 4py\), where \(p\) is the focal distance (the distance from vertex to focus).

Step 3: Determine the relationship between coordinates and the equation

Using the point (5, 3) that lies on the parabola, substitute into the equation \(x^2 = 4py\). Thus, we have \(5^2 = 4p(3)\).

Step 4: Solve for the focal distance (p)

Substitute the values into the equation: \(25 = 12p\). Solving for \(p\) gives \(p = \frac{25}{12}\).

Step 5: Interpret the result

The receiver should be placed at a distance of \(\frac{25}{12}\) feet from the vertex along the axis of symmetry of the parabola, which is approximately 2.08 feet.

Key Concepts

Parabola Equation

The equation for a parabola is a representation that allows us to identify and analyze its unique properties. A standard parabola that opens upward has the general form \( x^2 = 4py \). Here, \( x \) and \( y \) represent the coordinates on a Cartesian plane. The variable \( p \) represents the focal distance, which is the distance between the vertex of the parabola and its focal point.

Finding the parabola equation relies on using known dimensions of the parabola. In this case of the satellite antenna dish, the parabola's equation is based on the dimensions provided: the width across at the open end and the depth of the dish. This knowledge allows us to place a specific point on the parabola (such as (5, 3)) into the equation to solve for \( p \), helping to find the location of maximum intensity reception in the dish.

Focal Point

The focal point of a parabola is a vital concept, especially in the design and understanding of devices like satellite dishes. It's the point where sound waves, or other signals, reflect and converge after bouncing off the parabolic surface. In essence, the focal point acts as a concentrate zone for all rays reflecting from the parabola.

To find the focal point using the parabola equation, one must solve for \( p \) in the equation \( x^2 = 4py \). The focal distance \( p \) represents the distance from the vertex to the focus along the y-axis for an upward-opening parabola. By determining \( p \), we can effectively place the receiver at this point to ensure the optimal capture of signals or waves.

Satellite Antenna Dish

A satellite antenna dish uses the shape of a paraboloid to capture and concentrate signals. The parabolic shape of the dish ensures that all incoming parallel signals (like satellite transmissions) are reflected towards the focal point.

This focusing capability is crucial for efficient signal reception, making the placement of a receiver or sensor at the focal point of utmost importance. The design begins with understanding the parabola's equation and its geometrical qualities. By accurately calculating parameters such as focal distance, based on the dish’s dimensions, technicians and engineers can ensure optimal performance and signal strength.

Vertex of the Parabola

The vertex of a parabola is its most fundamental point, often used as a reference point in geometry and design. For a standard parabola, the vertex is commonly placed at the origin, \( (0,0) \), on the Cartesian plane, especially when analyzing geometric problems such as in a parabolic satellite dish setup.

The vertex is significant because it marks the midpoint of the opening of the parabola, providing symmetry and balance. Additionally, the vertex is directly related to the placement of the focus; the focal distance extends from this point. Knowing the distance from the vertex to the focal point assists in precise calibrations necessary for applications like antenna dishes, where the central positioning is symmetrical and intentional.