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I'm trying to solve this function, I've already gotten it to work when I set r but now I'm trying to do it when r remains a variable but this keeps on telling me NSolve doesn't have the methods available. I've tried using an approximation of ArcCosx and switching to Reduce and I'm not sure what I could try next?
l = 1000000/(r^2 Pi);
a = h/r;
NSolve[50000*1/l == r^2 ArcCos[1 - a] - (r - h) Sqrt[2 r h - h^2], h, Reals]
You could substitude h= \[Lambda] r and your equation becomes
eq = Simplify[(\[Pi] r^2)/20 == -(-h + r) Sqrt[-h^2 + 2 h r] + r^2 ArcCos[1 - h/r] /.h -> \[Lambda] r , r > 0]
(* 20 ((-1 + \[Lambda]) Sqrt[-(-2 + \[Lambda]) \[Lambda]] +
ArcCos[1 - \[Lambda]]) == \[Pi] *)
This equation can be solved
NSolve[eq, \[Lambda], Reals]
(* {{\[Lambda] -> 0.194616}}*)
