Three solid cubes of iron have edges measuring 6 cm, 8 cm and 10 cm respectively. They are melted and recast into a single solid cube. What is the length (in cm) of an edge of the new cube?

Introduction / Context:
This question checks your understanding of volume conservation when solid metal bodies are melted and recast. Cubes with different edge lengths are combined by melting and then recast into a single cube. The total volume of the metal stays the same, so the volume of the new cube equals the sum of the original volumes.

Given Data / Assumptions:
• Cube 1 edge length = 6 cm.
• Cube 2 edge length = 8 cm.
• Cube 3 edge length = 10 cm.
• All three cubes are made of the same material and are completely melted.
• No loss of material occurs during melting and recasting.

Concept / Approach:
Volume of a cube of edge a is a3 cubic units. The total volume before melting is the sum of the three individual volumes. If the edge of the new cube is L, then its volume is L3. Setting L3 equal to the total original volume allows solving for L by taking the cube root.

Step-by-Step Solution:
Step 1: Compute volume of cube 1: V1 = 63 = 216 cubic centimetres. Step 2: Compute volume of cube 2: V2 = 83 = 512 cubic centimetres. Step 3: Compute volume of cube 3: V3 = 103 = 1000 cubic centimetres. Step 4: Total volume Vtotal = V1 + V2 + V3 = 216 + 512 + 1000 = 1728 cubic centimetres. Step 5: Let edge length of new cube be L cm. Then L3 = 1728. Step 6: The cube of 12 is 123 = 1728, so L = 12 cm.

Verification / Alternative check:
Check the cube of 12 explicitly: 12 × 12 × 12 = 144 × 12 = 1728, which matches the total original volume. No other integer option given has a cube equal to 1728, so the result is unique and correct.

Why Other Options Are Wrong:
• 14 cm gives volume 143 = 2744, which is larger than 1728.
• 16 cm gives 163 = 4096, much larger than 1728.
• 18 cm gives 183 = 5832, far above the available metal volume.

Common Pitfalls:
Sometimes learners mistakenly add edge lengths instead of volumes or forget that volume is proportional to the cube of the edge length. Remember that melting and recasting preserves volume, not linear dimensions.

Final Answer:
The edge length of the new cube is 12 cm.