Answer:
6.06 inches square by 3.03 inches high
Step-by-step explanation:
An open-top box will have a minimum surface area when its lateral area is double its base area. That is, left-area + right-area = front-area + back area = bottom area.
With that in mind, the simplest way to compute the dimensions is to find the base edge length as the cube root of double the volume.
b = ∛(2×111 in³) ≈ 6.06 in
h = b/2 = 3.03 in . . . . . there are 4 faces; each is half the area of the base
The box is 6.06 inches square and 3.03 inches high.
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Solution using derivatives
Let b represent the edge length of the base. The volume is ...
V = b^2h
Then the height is ...
h = V/b^2
h = 111/b^2
and the total surface area is ...
A = b^2 + 4bh = b^2 +4·111/b
The area is maximized when the derivative is zero:
dA/db = 0 = 2b -444/b^2
Solving for b, we get ...
2b^3 = 444
b^3 = 222
b = ∛222 ≈ 6.06 . . . . . . as above
