Three solid cubes with edge lengths in the ratio 3 : 4 : 5 are melted to form a single cube whose space diagonal is 12√3 cm. Find the original edge lengths of the three cubes.

Introduction / Context:
Melt-and-recast problems conserve volume. The sum of the three cubes’ volumes equals the volume of the final cube, from which we find a scale factor for the original edges given the ratio 3:4:5.

Given Data / Assumptions:

• Edge ratios: 3 : 4 : 5 ⇒ edges = 3k, 4k, 5k.
• Final cube diagonal d = 12√3 cm ⇒ final edge a satisfies d = a√3 ⇒ a = 12 cm.
• No loss of material.

Concept / Approach:

• Volume of a cube = edge^3.
• Total initial volume = (3k)^3 + (4k)^3 + (5k)^3 = 27k^3 + 64k^3 + 125k^3 = 216k^3.
• Final volume = a^3 = 12^3 = 1728.
• Equate volumes and solve for k.

Step-by-Step Solution:

Verification / Alternative check:
Volumes: 6^3 + 8^3 + 10^3 = 216 + 512 + 1000 = 1728 = 12^3; matches exactly.

Why Other Options Are Wrong:

• 3,4,5 cm: Gives total volume 216; far too small.
• 9,12,15 cm: Gives total 9^3+12^3+15^3 = 3375; too large.
• None of these: Not applicable since 6,8,10 cm are correct.

Common Pitfalls:

• Using diagonal directly as volume without converting to edge.
• Arithmetic errors in summing cubes.

Final Answer:
6 cm, 8 cm, 10 cm