Three metal cubes with sides 5 cm, 4 cm, and 3 cm are melted and recast into a single cube. Find the edge length of the new cube.

Introduction / Context:Melt-and-recast problems conserve volume (neglecting loss). The sum of the original volumes equals the volume of the new cube. Therefore, find the total initial volume and take its cube root to get the new edge.

Given Data / Assumptions:

• Original cube edges: 5 cm, 4 cm, 3 cm.
• Volumes: 125, 64, and 27 cm^3 respectively.
• Conservation of volume: V_new = 125 + 64 + 27.

Concept / Approach:Compute V_total = 5^3 + 4^3 + 3^3 = 216 cm^3, then solve a^3 = 216 for a.

Step-by-Step Solution:V_total = 125 + 64 + 27 = 216 cm^3a = ∛216 = 6 cm

Verification / Alternative check:6^3 = 216; volumes balance exactly.

Why Other Options Are Wrong:8 cm would require 512 cm^3; 10 cm requires 1000 cm^3; both exceed the available metal volume.

Common Pitfalls:Adding side lengths instead of volumes; forgetting that volume scales as the cube of edge length.

Final Answer:6 cm