Finding d in Circle-Circle Intersection Equation

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Discussion Overview

The discussion revolves around solving for the distance \( d \) in the context of the intersection of two circles, given the area of overlap and the radius of the circles. The focus is on the mathematical equation that describes the area of intersection and the challenges in isolating \( d \) due to the presence of trigonometric and square root functions.

Discussion Character

  • Mathematical reasoning
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant presents the equation for the area of intersection of two circles and specifies the values for the area and radius, seeking to solve for \( d \).
  • Another participant suggests solving numerically as a potential approach.
  • Concerns are raised about the complexity of the equation due to the arccos and square root functions, making it difficult to isolate \( d \).
  • A suggestion is made to graph the left and right sides of the equation as functions of \( d \) to find the solution visually.
  • Participants agree that \( d \) must lie within the range of 0 to 24, and one participant expresses uncertainty about how to determine proximity to the correct value.
  • A later reply indicates that using graphing software suggests \( d \) is slightly less than 15.95, but this is presented as a finding rather than a conclusion.

Areas of Agreement / Disagreement

Participants generally agree on the numerical approach and the range for \( d \), but there is no consensus on a definitive method for isolating \( d \) from the equation. The discussion remains exploratory with various suggestions being made.

Contextual Notes

The discussion does not resolve the mathematical complexities involved in isolating \( d \) from the given equation, and assumptions about the behavior of the functions involved are not fully explored.

FysixFox
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Okay, so I've found out about how circle-circle intersection works ( http://mathworld.wolfram.com/Circle-CircleIntersection.html ). I'm working with the following knowledge:

  1. The area of the overlap is 100
  2. The two circles have the same radius, 12
  3. d is unknown

How would I solve for d in the following equation?

A = 2r2arccos(d/2r) - 0.5sqrt(4r2d2 - d4)

or, with the values put in:

100 = 288arccos(d/24) - 0.5sqrt(576d2 - d4)
 
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Solve numerically?
 
olivermsun said:
Solve numerically?

Numerically. Or at all. I don't know what to do when there's all this arccos and square root stuff everywhere aand I can't find out how to isolate d.
 
One way is to arrange your equation in any convenient way you like, and graph the left and right sides as functions of d.
 
olivermsun said:
One way is to arrange your equation in any convenient way you like, and graph the left and right sides as functions of d.


The problem is, there's only one possible answer for d. How would I even know if I got close? I mean, I know it's less than 24 and proooobably more than 12, but... that's just from thinking about the circles on a graph.
 
Well, for sure 0 ≤ d ≤ 24, so just try plotting over that interval first and see what happens.
 
olivermsun said:
Well, for sure 0 ≤ d ≤ 24, so just try plotting over that interval first and see what happens.

Hm, let's use a graphing software to save time... Aha! It seems that the equation works when d is slightly less than 15.95... thanks! :)
 

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