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.

Correct result:

x =  12 cm
y =  10 cm
z =  10 cm

Solution:

$$\begin{gathered} n=50 \\ a=4\text{ cm } \\ b=3\text{ cm } \\ c=2\text{ cm } \\ V=n\cdot a\cdot b\cdot c=50\cdot 4\cdot 3\cdot 2=1200{\text{cm}}^3 \\ 1200=2\times 2\times 2\times 2\times 3\times 5\times 5=2^4\times 3\times 5^2 \\ S=2\cdot (x_y+y_z+x_z)\dots min \\ {x}_1=\sqrt[3]{V}=\sqrt[3]{1200}\doteq 10.6266\text{ cm } \\ x=2\cdot 2\cdot 3=12\text{ cm} \end{gathered}$$

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

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

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