Introduction / Context:
Volume is conserved when a solid is melted and redrawn. Compute the sphere’s volume and equate it to the cylindrical wire’s volume to solve for the wire length. Unit consistency (mm to cm) is crucial.
Given Data / Assumptions:
- Sphere diameter = 18 cm ⇒ radius r_s = 9 cm
- Wire radius r_w = 0.2 mm = 0.02 cm
- V_sphere = (4/3) π r_s^3; V_wire = π r_w^2 * L
Concept / Approach:
- Set V_sphere = V_wire and solve for L.
Step-by-Step Solution:
V_sphere = (4/3)π * 9^3 = (4/3)π * 729 = 972π cm3.V_wire = π * (0.02)^2 * L = π * 0.0004 * L.Equate: 972π = 0.0004π L ⇒ L = 972 / 0.0004 = 2,430,000 cm = 24,300 m.
Verification / Alternative check:
Convert cm to m by dividing by 100: 2,430,000 / 100 = 24,300 m.
Why Other Options Are Wrong:
- 243 m / 2430 m / 24.3 m: Off by powers of 10 due to unit conversion or squaring radius mistakes.
Common Pitfalls:
- Using radius 0.2 cm instead of 0.2 mm = 0.02 cm.
- Forgetting to square the wire radius in π r^2 L.
Final Answer:
24300 m
Discussion & Comments