Since the cost is the same for all surface areas, we just want to find the box with minimum surface area that holds the given volume.
The surface area involved is one base and two short sides and two long sides, so
a = 2x^2 + 2xz + 4xz = 2x^2+6xz
But since z = 36000/2x^2 = 18000/x^2,
a(x) = 2x^2 + 108000/x
da/dx = 4x - 108000/x^2
= (4x^3-108000)/x^2
= 4(x^3-27000)/x^2
clearly, da/dx = 0 when x=30
Now you can just provide the dimensions and the cost.
An open box with a capacity of 36,000 cubic inches is to be twice as long as it is wide. The material for the box is $.10/sq ft. What are the dimensions of the least expensive box? How much does it cost?
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