Answer:
x = y = 26 cm; z = 13 cm
Step-by-step explanation:
The generic solution for the minimum material in an open-top box is that the box is square and half as high as it is wide. It is half a cube of twice the volume.
The dimensions of the square base are ...
∛(2·8788 cm³) = 26 cm
Then the height is half that, or 13 cm.
x = y = 26 cm; z = 13 cm
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If you need to see the development, you can use the method of Lagrange multipliers to find the minimum area for the given volume;
area = xy +2(xz +yz)
volume = xyz = 8788
We require each of the partial derivatives of L with respect to x, y, z, and λ to be zero.
L = xy +2(xz +yz) +λ(xyz -8788)
partial with respect to x: 0 = y+2z +λyz
partial with respect to y: 0 = x +2z +λxz
partial with respect to z: 0 = 2x+2y +λxy
partial with respect to λ: 0 = xyz -8788
From the first two equations, we have ...
λ = (y +2z)/(yz) = 1/z +2/y
λ = (x +2z)/(xz) = 1/z +2/x
Equating these expressions for λ, we find ...
1/z +2/y = 1/z +2/x ⇒ x = y
The third equation then tells us ...
λ = (2x +2y)/(xy) = 2/y +2/x
Comparing this to either of the first two expressions for λ, we see ...
1/z +2y = 2/x +2/y ⇒ z = x/2
This is the result we started the answer with:
x = y = 2z = ∛(2·8788 cm³)
