1. **Stating the problem:** We have a cube model with side length 7 cm. The cubes are sold in packages of 60 and 90 units. We want to find the packaging dimensions that minimize surface area and cost for these packages. 2. **Properties of the cube:** - All edges are equal: $s=7$ cm. - Each face is a square with area $s^2=49$ cm$^2$. - The cube has 6 faces, so surface area $=6s^2=294$ cm$^2$. - Volume of one cube $=s^3=343$ cm$^3$. 3. **Packaging problem:** We want to pack $n$ cubes ($n=60$ or $90$) in a rectangular box with dimensions $x,y,z$ (in cm). 4. **Constraints and goal:** - Volume of box $V=xyz \\geq n imes 343$ cm$^3$ (to fit all cubes). - Cubes are packed without gaps ideally, so $x,y,z$ are multiples of 7. - Minimize surface area $S=2(xy + yz + zx)$ to reduce packaging cost. - Cost $=10 \times S$. 5. **For Paket 60:** - Total volume needed $=60 \times 343=20580$ cm$^3$. - Let $x=7a$, $y=7b$, $z=7c$ where $a,b,c$ are integers. - Volume constraint: $7a \times 7b \times 7c = 343abc \geq 20580 \Rightarrow abc \geq \frac{20580}{343}=60$. - Find integer triples $(a,b,c)$ with $abc \geq 60$ minimizing surface area: $$S=2(xy + yz + zx)=2(49ab + 49bc + 49ac)=98(ab + bc + ac)$$ - Try factor triples of 60: - $(3,4,5)$: sum $=3\times4 + 4\times5 + 3\times5=12 + 20 + 15=47$ - $(2,5,6)$: sum $=10 + 30 + 12=52$ - $(3,3,7)$: sum $=9 + 21 + 21=51$ - Minimum sum is 47 for $(3,4,5)$. - So dimensions: $x=21$ cm, $y=28$ cm, $z=35$ cm. - Surface area $S=98 \times 47=4606$ cm$^2$. - Cost $=10 \times 4606=46060$. 6. **For Paket 90:** - Volume needed $=90 \times 343=30870$ cm$^3$. - $abc \geq \frac{30870}{343}=90$. - Factor triples of 90: - $(3,5,6)$: sum $=15 + 30 + 18=63$ - $(3,3,10)$: sum $=9 + 30 + 30=69$ - $(2,5,9)$: sum $=10 + 45 + 18=73$ - Minimum sum is 63 for $(3,5,6)$. - Dimensions: $x=21$ cm, $y=35$ cm, $z=42$ cm. - Surface area $S=98 \times 63=6174$ cm$^2$. - Cost $=10 \times 6174=61740$. 7. **General strategy:** - Express box dimensions as multiples of cube side. - Use volume constraint to find integer factors. - Minimize surface area by choosing factor triples minimizing $ab+bc+ac$. - This approach ensures minimal packaging material and cost. 8. **Applicability to non-cube shapes:** - The strategy works best when items pack perfectly in a grid. - For irregular shapes, packing efficiency and shape complexity affect optimal packaging. - The principle of minimizing surface area for given volume remains valid but requires more complex optimization.