Problem 94 - Intermediate Algebra

Question

Satellite Antennas. The cross section of the satellite antenna in the illustration is a parabola given by the equation \(y=\frac{1}{16} x^{2},\) with distances measured in feet. If the dish is 8 feet wide, how deep is it? (GRAPH NOT COPY)

Step-by-Step Solution

Answer

The dish is 1 foot deep.

Step 1: Understanding the Problem

We need to find the depth of a parabolic satellite dish given its width. The cross-section of the dish is described by the equation \(y = \frac{1}{16}x^2\), and the dish width is 8 feet.

Step 2: Identifying Key Points

Since the dish is 8 feet wide, it spans from \(x = -4\) to \(x = 4\) (since 8 feet is 4 feet to the left and 4 feet to the right of the vertex).

Step 3: Finding the Depth

Substitute \(x = 4\) into the equation \(y = \frac{1}{16}x^2\) to find the depth of the dish.\[ y = \frac{1}{16} \cdot 4^2 \]

Step 4: Solving the Equation

Calculate the value of \(y\) by substituting \(x = 4\):\[ y = \frac{1}{16} \cdot 16 = 1 \]Thus, the depth of the dish is 1 foot.

Key Concepts

Understanding Satellite Antennas

Satellite antennas are fascinating structures, integral components in modern telecommunications systems. They play a crucial role in receiving and transmitting signals across massive distances, often from satellites orbiting high above the Earth's surface. The key to their efficient functioning lies in their shape and size.

• Purpose: Satellite antennas focus signals to a point for clearer reception and transmission. They convert electromagnetic signals into electrical signals and vice versa.
• Design: These antennas typically have a parabolic shape, allowing them to capture and focus radio waves. The parabolic shape ensures that all signals reflecting off the surface are directed towards a common focal point.
• Material: They are usually constructed from lightweight materials, such as aluminum or fiberglass, to withstand various weather conditions.

The parabolic design of satellite antennas is not arbitrary; it is a calculated choice that serves to maximize performance and efficiency. This design choice directly relates to the core geometry of parabolas.

The Mathematical Magic of Parabolas

A parabola is a symmetric, open curve. It is the set of all points that are equidistant from a point called the focus and a line called the directrix. In the context of satellite antennas, the parabolic shape allows signals to reflect off the dish surface towards the focus.

• Equation Form: The typical equation of a parabola can be written as \( y = ax^2 \), where the "a" value determines the parabola's "openness."
• Focus: Signals collected by the antenna focus at this point, and from here, they are transmitted or received.
• Symmetry: The vertical axis of symmetry makes it easy to calculate specific points, like the depth of a dish, given a certain width.

When dealing with satellite dishes, the parabolic shape ensures that all incoming signals are concentrated precisely at the receiver located at the focus, minimizing signal loss and maximizing clarity.

Depth Calculation in Parabolic Antennas

Understanding how to calculate the depth of a parabolic antenna is essential for determining how it captures signals. The dish's depth relates directly to its ability to focus incoming signals. In our exercise, we find the depth using the given width.

• Width Determination: By recognizing that half of the width extends from the vertex to each side, we have the total width spanning from \(x = -4\) to \(x = 4\).
• Substitution Method: To solve for depth, substitute the outermost x-value (which is \(x = 4\)) into the equation \(y = \frac{1}{16}x^2\).
• Solution: Calculate \(y = \frac{1}{16} \times 16 = 1\), giving a depth of 1 foot.

Each step in the calculation clarifies how the symmetry and properties of the parabola help you find precise measurements, crucial for designing effective satellite dishes.