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Optimal box

25.4.3
Optimal box
Fig. 1
Optimal box with bottom
B* Optimum width h* Optimum height C Aspect ratio
Eqn. 1
\require{color}\definecolor{myred}{RGB}{255,0,0} B^\ast=\sqrt[3]{\left(\frac VC+\frac V{C^{\color{myred}2}}\right)}
Eqn. 2
\require{color}\definecolor{myred}{RGB}{255,0,0} h=\frac V{C\cdot B^{\color{myred}2}}
Eqn. 3
\require{color}\definecolor{myred}{RGB}{255,0,0} A=B^{\color{myred}2}\cdot C+2\cdot B\cdot h\cdot\left(1+C\right)
Eqn. 4
\require{color}\definecolor{myred}{RGB}{255,0,0} q=\frac{A^\ast}{A_{\color{myred}ref}}-1
Optimum widthB*=103.57mm 
Optimal heighth*=62.14mm 
Optimal areaA*=48,274.5mm2 
Heighthref=82.3mm 
SurfaceAref=49,187mm2 
Percent lossq=-1.9% 
VolumeVol = 1L
ConstantC = 1.5
Reference widthBref = 90mm
Calc 1

Example 1 You are looking for a rectangular container with a filling volume of 1 L. One side of the base area (C x B) is 1.5 x larger than the desired dimension B --> C = 1.5. The optimal dimension B is calculated to be 103.5 mm and the optimal height to be 62.1 mm.

Due to the available installation space, dimension B is set to 90 mm, which results in a container height of 82.3 mm. This leads to an area loss of 1.9%.

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