Working with Surface of Revolution of Inverse Square

In summary, the problem is to determine the location of an unknown gamma radiation source. The relationship between radiation exposure rate and distance from the source is defined by the equation y = k / x^2, where y is the dose rate, x is the distance from the source, and k is a constant. To apply this in three dimensions, the equation needs to be rotated around the y-axis to form a surface of revolution. The equation for this surface is x^2 + z^2 = 1 / y^2. The highest y-reading on the x-z plane may not always be at the x-z point (0,0) as the curve can be rotated around any line parallel to the y-axis. The goal is
  • #1
natai
2
0
The problem below is actually in reference to determining the location of a unknown gamma radiation source. However, I believe the solution lies with relatively simple calculus.

First, the equation that defines the relationship between the radiation exposure rate and the distance from the source is defined by:
y = k / x^2
Where y is the dose rate, x is the distance from the source, and k is a constant which may vary depending on the specific situation and source.

Next, this needs to be applied in three dimensions (x,y,z). So I believe it needs to be rotated around the y-axis to form a surface of revolution. If I remember correctly, the equation for such a surface should look something like:
x^2 + z^2 = 1 / y^2
On this curve your distance from the y-axis on the x-z plane would be equivalent to your distance frlom the radiation source, and the y-coord corresponding to each (x,z) would be the dose rate at that distance.

Think of it this way. At point (x,y,z) x and z are like latitude and longitude while y is the radiation dose rate.

This curve could be rotated around any line parallel to the y-axis, so the highest y reading would not always be at x-z point (0,0).

Here is what I am trying to accomplish:
If I have multiple (x,y,z) points where x and z are latitude and longitude (or just points on the x-z plane measured in feet) and y is the radiation rate, I want to be able to locate the highest y-reading on the x-z plane. In other words, if I know a radiation reading at two or three points and the latitude and longitude at those points, I want to be able to locate latitude and longitude of the radiation source.

Any ideas?
 
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  • #2
I am not sure whether you have one question or two questions:

1. "If I have multiple (x,y,z) points where x and z are latitude and longitude (or just points on the x-z plane measured in feet) and y is the radiation rate, I want to be able to locate the highest y-reading on the x-z plane."
2. "if I know a radiation reading at two or three points and the latitude and longitude at those points, I want to be able to locate latitude and longitude of the radiation source."
 
  • #3
Actually, those questions are one and the same as the radiation source will be the highest reading.
 
  • #4
I am interpreting your questions. Let's say you estimated a least squares regression of the form y = a + b x + c z + u where a, b, c are parameters and u is the random error. Now for any (x, z) pair, you can estimate the corresponding y. However, one of your expressions reads "I want to be able to locate latitude and longitude of the radiation source." Which makes me think that maybe you wish to go the other way around: from y to (x, z), which is not insolvable, but it is somewhat more complicated.
 

1. What is a surface of revolution of inverse square?

A surface of revolution of inverse square is a three-dimensional surface that is created by revolving a two-dimensional curve around a central axis, where the distance from the curve to the axis is inversely proportional to the square of the distance from the axis. This type of surface is commonly found in physics and engineering, particularly in the study of gravity and electromagnetism.

2. How is a surface of revolution of inverse square used in scientific research?

A surface of revolution of inverse square is used in scientific research to model and understand the behavior of physical phenomena, such as the gravitational or electrical fields around a point source. It is also used in the design and analysis of various engineering structures, such as antennas and reflectors.

3. Can a surface of revolution of inverse square be described mathematically?

Yes, a surface of revolution of inverse square can be described mathematically using equations such as the inverse square law, which states that the intensity of a physical quantity is inversely proportional to the square of the distance from the source. Other mathematical equations, such as integrals and differential equations, can also be used to describe these surfaces.

4. What are some real-life examples of a surface of revolution of inverse square?

There are many real-life examples of a surface of revolution of inverse square, including the shape of a satellite dish, the electric field around a point charge, and the gravitational field around a planet or star. These surfaces also appear in nature, such as in the shape of a water droplet or the structure of a galaxy.

5. What are some applications of a surface of revolution of inverse square in technology?

A surface of revolution of inverse square has many applications in technology, including the design of parabolic reflectors in satellite dishes and telescopes, the creation of magnetic and electric fields in particle accelerators, and the analysis of antenna patterns in wireless communication systems. It is also used in the design and optimization of various engineering structures, such as mirrors, lenses, and reflectors.

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