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The surface area of the box as a function of x will be f(x) = x² + 20000/x² where x is the length of the square base's edge.
Here it is given that the box has a square base.
This implies that length = breadth
Now, the volume of a box = lbh = 5000 cu in
where l is the length of the box. b is the breadth of the box, and h is the height of the box.
or, x²h = 5000
or, h = 5000/x²
Let the surface area function be f(x) where x is the length of the edge of the square base.
Since the box is open, the surface area will be
lb + 2bh + 2lh
= x² + 10000/x² + 10000/x
= x² + 20000/x²
Hence the surface area as a function of x will be
f(x) = x² + 20000/x²
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