I have a problem for my vector calculus class where we have to build a storage tank that is the shape of a surface of revolution obtained by rotating

x^2 + a^2 y^2 = 1, x > 0, around the y-axis. Given a fixed surface area S the problem is to find the best value of a to maximize the volume.

I'm honestly so fried with abstract classes that I'm having trouble approaching this problem. Any help would be greatly appreciated.