Problem 44 - Algebra and Trigonometry with Analytic Geometry

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

The receiver should be placed approximately 2.08 feet from the vertex of the dish.

Step 1: Identify the components of the parabola

The shape of the satellite dish is a paraboloid, which is a 3D extension of a 2D parabola. We consider a cross-section of the dish as a parabola with its vertex at the origin. Let the parabola be described by the equation \( y = ax^2 \), where \( y \) represents the depth and \( x \) the horizontal distance from the center.

Step 2: Determine the parabola equation parameters

The parabola is 10 feet across at the open end, implying that the distance from the vertex to the edge is 5 feet on either side. It is also 3 feet deep. Substitute the points \((5, 3)\) into the parabola equation to find the value of \(a\), giving us \(3 = a(5^2)\). Solving for \(a\) yields \(a = \frac{3}{25}\). Hence, the parabola's equation is \(y = \frac{3}{25}x^2\).

Step 3: Find the focus of the parabola

The parabola equation is rewritten in the standard form \(4p(y) = x^2\). Compare it to \( y = \frac{3}{25}x^2 \), which implies \( 4p = \frac{25}{3} \). Solving for \(p\), \( p = \frac{25}{12} \). This value indicates the focal distance from the vertex along the y-axis.

Step 4: Calculate the focal distance

The focal point is located \( p = \frac{25}{12} \approx 2.0833 \) feet from the vertex along the axis of symmetry in the parabola (positive y-direction from the vertex). This is the location where the receiver should be placed to receive the maximum intensity of sound waves.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. It is defined as the set of all points in a plane that are equidistant from a fixed point known as the "focus" and a line called the "directrix." In simpler terms, a parabola is the path followed by a point that moves in a plane in such a way that its distance from a fixed point (the focus) is always equal to its distance from a fixed line (the directrix).

Parabolas have several important properties:

• They are symmetric with respect to a vertical axis called the axis of symmetry.
• The lowest or highest point, depending on the direction the parabola opens, is called the vertex.
• Real-world applications of parabolas include satellite dishes, car headlights, and bridge arches.

Understanding the basic geometric properties of parabolas helps in creating structures like satellite dishes, where precise mathematical calculations are crucial.

Satellite Dish

A satellite dish is a type of parabolic reflector used to receive or transmit information to a satellite. The dish is a parabolic-shaped surface that collects signals and focuses them to a specific point. This design is crucial for satellite communication as it ensures that the dish captures the maximum possible signal strength and minimizes interference.

Satellite dishes are typically designed as paraboloids. This means they extend the principles of a two-dimensional parabola into a three-dimensional shape, creating a surface called a paraboloid. Here's why the parabolic shape is ideal:

• It allows signals, which travel in parallel beams, to be focused at a single point, the focus.
• This focused point is where the receiver is placed, ensuring optimal signal capture.
• The shape also helps to deflect unwanted noise and interference, improving the clarity of received signals.

The precision in constructing the parabolic shape is key to ensuring the satellite dish functions effectively.

Focus of a Parabola

The focus of a parabola is one of its defining elements. It is a specific point inside the curve from which distances to any point on the parabola are equal to the distances from the point to the directrix.

To understand its importance, consider the following:

• The focus is crucial as it determines the direction and width of the parabola.
• In satellite dishes, the focus is the point where signals are concentrated and where the receiver should be placed.
• Mathematically, the distance from the vertex to the focus is called the "focal length," represented often by the letter "p." The formula for the distance is derived from the parabola's equation: if the equation is in the form \(4p(y) = x^2\), then the focal length is \(p\).
• The position of the focus is critical for achieving maximum signal strength in applications involving sound or light waves, such as satellite dishes or optical devices.

For any parabola, precisely locating and aligning the focus ensures the system's efficiency, whether in communication technology or architectural developments.

Equation of a Parabola

The equation of a parabola is a mathematical expression that describes its shape and position. In the context of satellite dishes, this equation is crucial for determining how signals will be collected and focused.

The standard form of a parabola's equation, especially when aligned with the vertical y-axis, is \(y = ax^2\). Here, "a" determines the parabola's openness and direction:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

From the given problem:

• The dish is 10 feet across, which implies \(x = 5\) when \(y = 3\). By substituting these values into the equation and solving for "a," we find \(a = rac{3}{25}\).
• Thus, the parabola's equation for the satellite dish is \( y = \frac{3}{25}x^2\).

This parabola equation helps locate the focus, further guiding where the receiver should be placed for optimal performance. Understanding both the formation and the role of the parabola equation is essential in fields ranging from architecture to signal processing.