Question

A satellite dish is shaped like a paraboloid of revolution. 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?

Short answer

The receiver should be placed 2.25 feet above the base at the center.

Step-by-step solution

Understand the problem

The shape of the satellite dish is a paraboloid, which is a 3D parabola. We need to find the focus of this parabolic shape where the receiver should be placed.

Set up the equation of the parabola

The cross-section of the parabolic dish can be described by the equation \( y = ax^2 \), where the vertex is at the origin (0,0), the x-axis is horizontal, and the y-axis is vertical. The dish is 4 feet deep and has an opening of 12 feet, meaning the parabolic equation goes through the point (6,4) since half of the opening width (12 feet) is 6 feet.

Solve for the coefficient 'a'

Substitute the point (6,4) into the parabola equation: \( 4 = a(6)^2 \). Solve for \( a \):\[ 4 = 36a \] \[ a = \frac{1}{9} \].

Identify the formula for focus

The focus of a parabola in the form \( y = ax^2 \) is given by \( (0, \frac{1}{4a}) \).

Calculate the position of the focus

Substitute \( a = \frac{1}{9} \) into the focus formula: \( \frac{1}{4 \times \frac{1}{9}} = \frac{9}{4} \). Therefore, the focus is at \( (0, \frac{9}{4}) \) or (0, 2.25).

Conclusion

The receiver should be placed at the focus, which is 2.25 feet above the base of the dish at its center.

Key concepts

Focus of a parabola

The focus of a parabola plays a crucial role in its geometric properties, particularly in applications such as satellite dishes and telescopes. For a 2D parabola, the focus is a single point that every point on the parabola is equidistant from together with a specific directrix line.
The focus is not just a mathematical curiosity; it is essential in defining how light or signals are reflected. Specifically, for parabolic dishes, any signal or wave that comes parallel to the axis of the parabola will reflect off the surface and converge at the focus. This is why receivers are placed at the focus of satellite dishes, allowing them to collect signals efficiently.

• The point is inside the curve for a parabola opening upwards or downwards.
• The standard form equation to find the focus when the parabola's vertex is at the origin and in the form \(y = ax^2\) is \((0, \frac{1}{4a})\).
• This calculation was used in the exercise to position the receiver precisely at 2.25 feet above the dish's base.

Understanding the focus helps in engineering applications where precision is critical, such as the case of aligning receivers.

Equation of a parabola

Parabolas have a distinctive mathematical representation, making them fundamental in geometry and algebra. To describe a parabola mathematically, you need its equation. The standard form of a parabola's equation is typically \(y = ax^2\), which assumes the vertex is at the origin. This represents parabolas that open upwards or downwards.

In real-world problems like the satellite dish from our exercise, the understanding of this equation allows precise calculation of parameters. To determine the specific equation for a given parabola, such as a dish, we often rely on known points through which the parabola passes. For the satellite dish, these points included the depth and width measurements (i.e., passing through (6,4)). Substituting these into the parabola equation enables us to solve for the coefficient \(a\).

• The coefficient \(a\) influences how "steep" or "flat" the parabola appears. Here \(a = \frac{1}{9}\), indicating a relatively gentle slope, suitable for focusing signals.
• The point calculations using the parabola help design efficient satellite dishes by knowing exactly where the focus will lie.
• Mathematically, this efficiency in design stems from solving equations and evaluating how parabolic shapes behave.

This precise mathematical form helps engineers and scientists align satellite dishes precisely with their targets, ensuring maximum signal reception.

Parabolic dish

Parabolic dishes, like the one described in the exercise, creatively apply the geometric properties of parabolas to real-world uses. These dishes are essentially 3D extensions of the 2D parabolic shape, forming what is known as a paraboloid.

The magic of a parabolic dish, such as a satellite dish, lies in its ability to focus all incoming signals or beams that are parallel to its axis onto a single point, the focus. This unique property is a direct consequence of the mathematical nature of parabolas. Here's how it works in practice:

• The dish collects a large amount of signal and concentrates it at the receiver located at the focus, maximizing clarity and signal strength.
• Its shape ensures even signals coming in at shallow angles will be directed to the focus.
• Size and alignment matter in a parabolic dish, making the calculations for focus position crucial for optimum performance.

When designers engineer these dishes, they consider the parabolic form to maximize efficiency and signal capture. The specific calculations made (like those in the solution) assist in designing effective dishes that meet communication needs by positioning the receiver precisely at the computed focus.