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:
π r^2 L = π (r^2 / 9) * L_new.Cancel π r^2: L = (1/9) L_new ⇒ L_new = 9 L.
Verification / Alternative check:
Area scales with r^2; decreasing radius by factor 3 decreases area by 9; length must grow by 9 to keep volume constant.
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
Discussion & Comments