Finding d in Circle-Circle Intersection Equation

[FysixFox]

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)

 

[olivermsun]

Solve numerically?

 

[FysixFox]

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.

 

[olivermsun]

One way is to arrange your equation in any convenient way you like, and graph the left and right sides as functions of d.

 

[FysixFox]

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.

 

[olivermsun]

Well, for sure 0 ≤ d ≤ 24, so just try plotting over that interval first and see what happens.

 

[FysixFox]

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! :)