Problem 50 - Algebra and Trigonometry

Question

A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus \(F\). The reflector is 1 ft deep and 20 ft wide from rim to rim (see the figure). How far is the receiver from the vertex of the parabolic reflector?

Step-by-Step Solution

Answer

The receiver is 25 ft from the vertex of the parabolic reflector.

Step 1: Understand the Parabola Equation

A parabola has the standard form equation \( y = ax^2 \). The vertex, which is at \( (0,0) \), will make it easier to find the focal point.

Step 2: Define the Known Dimensions

The parabola is symmetric about the y-axis with its width of 20 ft, meaning that at its widest point, \( x = 10 \) on either side of the y-axis. The depth at x=0 is 1 ft, so the point (10, 1) lies on the parabola.

Step 3: Use the Known Point to Find 'a'

Substitute the known point (10, 1) into the equation \( y = ax^2 \) to find 'a':\[ 1 = a(10)^2 \]\[ 1 = 100a \]\[ a = \frac{1}{100} \]

Step 4: Find the Focus Distance from the Vertex

The formula for the vertex form of a parabola with a vertical axis is \( y = \frac{1}{4p}x^2 \), where \( p \) is the distance from the vertex to the focus. Since \( a = \frac{1}{100} \), we equate it to \( \frac{1}{4p} \):\[ \frac{1}{100} = \frac{1}{4p} \]Solve for \( p \):\[ 4p = 100 \]\[ p = 25 \]Thus, the receiver is 25 ft away from the vertex.

Key Concepts

Parabolic Reflector

Parabolic reflectors are widely used in structures like satellite dishes and headlights. They have a distinctive shape defined by the parabola. This shape is essential because of its optical properties. A parabolic reflector can direct all incoming rays parallel to its axis towards a single focal point.

Important aspects to understand include:

• The shape of the reflector is a "cross-section" of a parabola.
• The edge of the reflector is typically widest at the "rim."
• The focus is where the reflected light (or signal) collects, optimizing efficiency.

These characteristics make parabolic reflectors very effective for strengthening signals in satellite dishes by pointing them all to the receiver at the focus.

Focus of a Parabola

The focus of a parabola is a key point that does remarkable things. It's the point where parallel rays, when reflected off a parabolic surface, will converge and meet.

Here's why the focus is important:

• A parabolic dish will "catch" signals and direct them toward the focus, allowing a small receiver at the focal point to pick up signals with strength and clarity.
• The focal point is characterized mathematically as being located a specific distance from the vertex—a measure called the "focal distance."

In the satellite dish problem, the focus is where the receiver is placed to maximize signal reception from the satellite.

Vertex Form of a Parabola

The vertex form of a parabola is incredibly useful. It allows us to quickly determine important characteristics like the focal distance from its equation.

The general formula is:
\[ y = a(x-h)^2 + k \]
Here,

• \((h, k)\) represents the vertex of the parabola.
• The value of \(a\) helps us understand how "wide" or "narrow" the parabola is.

The standard form often presented as \( y = ax^2 \), when simplified back, can reveal information like whether the parabola opens upwards or downwards. In applications like the parabolic reflector, it's crucial for determining the position of the focus relative to the vertex.

Focal Distance

Focal distance is the length between a parabola's vertex and its focus. This measure is essential in applications that require precise control of waves or signals, such as antennas or reflectors.

In mathematical terms, the focal distance \( p \) can be found using the vertex form formula:
\[ a = \frac{1}{4p} \]
In the case of the parabolic reflector exercise, this concept helped determine how far the receiver (focus) is from the vertex.

Key points include:

• The relation between \(a\) and \(p\) lets us calculate the position of the focus once the parabola's equation is known.
• The larger the focal distance, the further away the focus is from the vertex, impacting how the parabola gathers and focuses incoming signals.

Understanding focal distance allows for the optimization of technology relying on parabolic shapes.