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 answer:

x =  12 cm
y =  10 cm
z =  10 cm

Step-by-step explanation:

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

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

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

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