Answer :
The minimum dimensions to minimize the amount of material is 15.87 x 15.87 x 15.87 inch.
We need to know about the minimum function to solve this problem. The minimum function can be defined as the minimum value of the variable. It can be calculated by the derivative of the function. The minimum function can be written as
f'(s) = 0
where f'(s) is the derivative function
From the question above, the given parameter is
V = 4000 inch³
The cubic volume can be written as
s² x h = 4000
where s is the length of the base area and h is the height of the box.
the minimum value of the box refer to its surface area, hence
f(s) = 4sh + 2s²
f'(s) = 4h + 4s
4h + 4s = 0
h = s
It means that the length of the base area and the height must have the same dimension. The minimum volume should have a cube shape
V = s³
s³ = 4000
s = 15.87 inch
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