Minimum surface

Find the length, breadth, and height of the cuboid shaped box with a minimum surface area, into which 50 cuboid shaped blocks, each with length, breadth and height equal to 4 cm, 3 cm and 2 cm respectively can be packed.

Result

x =  12 cm
y =  10 cm
z =  10 cm

Solution:

$n=50 \ \\ a=4 \ \text{cm} \ \\ b=3 \ \text{cm} \ \\ c=2 \ \text{cm} \ \\ \ \\ V=n \cdot \ a \cdot \ b \cdot \ c=50 \cdot \ 4 \cdot \ 3 \cdot \ 2=1200 \ \text{cm}^3 \ \\ 1200=2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5=2^4 \times 3 \times 5^2 \ \\ \ \\ S=2 \cdot \ (xy+yz+xz) ... min \ \\ \ \\ x_{ 1 }=\sqrt[3]{ V}=\sqrt[3]{ 1200 } \doteq 10.6266 \ \text{cm} \ \\ \ \\ x=2 \cdot \ 2 \cdot \ 3=12 \ \text{cm}$

$y=5 \cdot \ 2=10 \ \text{cm}$

$z=V/ (x \cdot \ y)=1200/ (12 \cdot \ 10)=10 \ \text{cm}$

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