Wire length change with radius scaled: A wire is stretched so that its radius becomes one-third of the original while volume remains constant. By what factor does its length increase?

Introduction / Context:For a wire modeled as a cylinder, volume V = π r^2 L. If radius changes but volume is fixed, length must adjust inversely with r^2. This is a pure proportionality problem.

Given Data / Assumptions:

• Initial radius r, length L; final radius r/3; volume constant.

Concept / Approach:

• V_initial = π r^2 L, V_final = π (r/3)^2 L_new.
• Equate V_initial = V_final and solve for L_new / L.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 6 times / 2 times / 1 time: Do not satisfy V constant with r → r/3.

Common Pitfalls:

• Thinking length scales inversely with r, not r^2.
• Forgetting cylindrical volume formula.

Final Answer:

9 times