Context
I'm in year 12 of school, and for our curriculum, we have to write a 12-page thesis on any mathematical subject. Since I picked maths for Higher Level the maths has to be commensurate for that level (and sometimes a tad bit above).
The problem I am facing
For my thesis, I'm working with Lagrange multipliers for one of the parts but every single time I try to solve the equations which I got from rearranging so that they are equal to the multiplier phi, I'm either getting a complex volume (function 1 = function 2) or an unsolvable function (function 2 = function 3).
My work:
Three functions using Lagrange multipliers
Setting Eq. 1 and Eq. 2 (accidently wrote eq. 3 on paper) equal to each other to solve for them V by substituting h= (3V)/(ab)
Equating (2) and (3)
$$ \sqrt{h^2 + \frac{a^2}{h} }+ \frac{\sqrt{h^2 + \frac{a^2}{h}}}{\sqrt{h^2 + \frac{b^2}{h}}} = 4ab$$
Reference
Any and all help would be much appreciated. I'm using this thesis as a reference (bottom part is about Lagrange's multipliers).