This is very strange. You are doing Calculus yet you say "However, the pi is scaring me. Pi is a constant and so far I have only learnt doing it with variables." Have none of the problems you have done involve "2" or "3"? Those are also constants. Even if you had a problem with, say , "xyz", there is a "1" implied that is a constant.
You then have " " and " ". Where did those come from? You say you are using the "Lagrange multiplier method" but if so where is the multiplier? Yes, 2rh= 4r implies h= 2r but did you notice that immediately implies r= 0 or r= 2 (which is not a solution)? And your third "equation", is not even an equation!
The idea of the "lagrange Multiplier Method" is that any maximum or minimum of F(x,y,z) subject to the constraint G(x,y,z)= constant occurs where for some constant .
Here, F is , the volume function, and G is , the total surface area.
, where I have arbitrarily written for a unit vector in the "r direction" and for a unit vector in the "h direction".
Setting the components equal we have the two equations
Those, together with the equation , give three equations to solve for r, h, and .
Since it is not necessary to find to solve this problem, it is often simplest to eliminate by dividing one equation by another:
which reduces to
Solve that, together with .