Let the side of the base be of length x and the height y.

The volume is then x^2y = 250.

Top + Bottom = x^2 + x^2 = 2x^2 square meters, at 2 dollars each, this gives a top+bottom cost of 4x^2.

Sides' area = 4xy, at one dollar per square meter, this gives a cost of 4xy dollars

Total Cost =

Restriction:

Restriction: Total Cost

Total cost =

Optimal cost:

C(5) is a minimum because at x=5.

The minimum cost is $300. So then your answer would be no, you cannot build this box with less than 300 dollars, but you can build it with exactly 300 dollars.