Question

The material for the base of a box will cost three times as much as the material for the sides and top of the box. The box must have a volume of 200 meters cubed. Find the most efficient way to built this box.

Answer 1

Having seen and done hundreds of this type of question, I am pretty sure that the base is a square.
Else you don't have enough information to solve the question.

let the side of the base be x
let its height be y
so (x^2)y = 200 ---> y = 200/x^2

Cost = 3 money units(base) + 1 money unit(4 sides) + 1 money unit(top)

Cost = 3x^2 + 4(xy) + x^2
C = 4x^2 + 4xy
= 4x^2 + 4x(200/x^2)
= 4x^2 + 800/x
C' = 8x - 800/x^2 = 0 for a min of C
8x = 800/x^2
x^3 = 100
x = 4.64 and y = 9.28

PS, if the volume had been 2000 m^3 we would have had nice integer values.
Are you sure you did not make a typo?