9514 1404 393
Answer:
x = ∛4 ≈ 1.587 m
y = (∛4)/2 ≈ 0.794 m
Explanation:
Short answer:
An open-top box will use minimum material when it has the shape of half a cube. That is, the x-dimensions will be ...
x = ∛(2·2 m³) = ∛4 m
The y-dimensions will be half that:
y = x/2 = (∛4)/2 m
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Long answer:
The volume is ...
V = x^2·y
so the y-dimension is ...
y = V/x^2
The area of the sides and bottom will be ...
A = 4xy + x^2
A = 4x(2/x^2) +x^2 = 8/x +x^2
The area is minimized when the derivative of this is zero.
A' = 0 = -8/x^2 +2x
x^3 = 4 . . . . . . . divide by 2 and rearrange
x = ∛4 . . . . . . . . cube root
y = 2/x^2 = 2∛4/4
y = (∛4)/2
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Additional comment
If you follow the numbers through, you see that the value under the cube-root radical is twice the volume. This is the result we used in the "short answer."