# Finding roots of equation

I have the next equation:

I wrote it in mathematica and I obtained

f[h_] = 8 - 5 (-(2 - h) Sqrt[4 h - h^2] + 4 Sec[(2 - h)/2])

Now I have to solve it, finding the h. I know that h is between 0<=h<=4 by the Sqrt equation but I don't know how to find the real root. I tried with NSolve or FindRoot but it didn't work.

• you know the notation cos^-1 usually denotes inverse cosine not 1/cos..(sec) Commented Apr 25, 2018 at 1:21
• indeed changing Sec to ArcCos you should get a solution around h=.74 Commented Apr 25, 2018 at 1:29

EDIT: Corrected eqn for missing L

eqn = V == (r^2 ArcCos[(r - h)/r] - (r - h) Sqrt[2 r h - h^2])L;

const = {r -> 2, L -> 5, V -> 8};

The exact solution for h is a Root object

sol = Solve[eqn /. const, h, Reals][[1]]

{h -> Root[{-(8/5) + 4 ArcCos[(2 - #1)/2] -
2 Sqrt[-(-4 + #1) #1] + #1 Sqrt[-(-4 + #1) #1] &,
0.74001521805594051394}]}

The numeric approximation is

sol // N

(* {h -> 0.740015} *)

Verifying that the solution satisfies the equation

eqn /. const /. sol // FullSimplify

(* True *)

or

eqn /. const /. N[sol]

(* True *)

Plot[Evaluate[eqn /. const], {h, 0, 4},
Epilog -> {Red, PointSize[Medium], Point[{h /. sol, 0}]}]