The dimensions of a piece of iron in the shape of a cuboid are 270 cm $\times$ 100 cm $\times$ 64 cm. If it is melted and recast into a cube, then the surface area of the cube will be

### Concept & Logic When a solid is melted and recast into another solid, the volume remains constant. Therefore, the volume of the new cube will equal the volume of the original cuboid. After finding the cube's side length from its volume, calculate its surface area. $$Volume_{cuboid} = L \times B \times H$$ $$Volume_{cube} = a^3$$ $$Surface Area_{cube} = 6a^2$$ ### Step-by-Step Solution 1. **Equate Volumes:** $a^3 = 270 \times 100 \times 64$ 2. **Find Side Length ($a$):** Instead of multiplying out to a massive number, factorize to find the cube root easily. $a^3 = (27 \times 10) \times 100 \times 64$ $a^3 = 27 \times 1000 \times 64$ $a^3 = 3^3 \times 10^3 \times 4^3$ Taking the cube root: $a = 3 \times 10 \times 4 = 120$ cm. 3. **Calculate Surface Area:** $Surface Area = 6a^2$ $Surface Area = 6 \times (120)^2$ $Surface Area = 6 \times 14400$ $Surface Area = 86400$ cm$^2$. ### Exam Strategy & Shortcut Factoring before multiplying is crucial here. Breaking down $270 \times 100 \times 64$ into perfect cubes ($27, 1000, 64$) makes finding the cube root trivial ($3 \times 10 \times 4$). ### Common Pitfall Multiplying the entire volume out to $1,728,000$ and then struggling to find the cube root manually. Always look for factors when dealing with roots. ### Final Answer Therefore, the correct answer is **86400 cm$^2$**.