Introduction / Context:
When a solid is reshaped without loss, its volume remains constant. Here, a copper sphere is redrawn into a thin cylindrical wire; therefore, sphere volume = wire volume. The task is to compute the wire length from these equal volumes.
Given Data / Assumptions:
- Sphere diameter = 36 cm ⇒ radius r_s = 18 cm.
- Wire diameter = 4 mm = 0.4 cm ⇒ radius r_w = 0.2 cm.
- No loss of material; volumes are equal.
- Answer required in metres.
Concept / Approach:
- Volume of sphere: V_s = (4/3)*π*r_s^3.
- Volume of cylinder (wire): V_w = π*r_w^2*L.
- Set V_s = V_w and solve for length L, then convert cm → m.
Step-by-Step Solution:
V_s = (4/3)*π*(18)^3 cm^3V_w = π*(0.2)^2*L cm^3Equate: (4/3)*π*18^3 = π*0.2^2*LCancel π: (4/3)*18^3 = 0.04*LCompute 18^3 = 5832 ⇒ (4/3)*5832 = 77760.04*L = 7776 ⇒ L = 7776 / 0.04 = 194400 cmConvert to metres: L = 194400 / 100 = 1944 m
Verification / Alternative check:
Quick ratio check: Length ∝ r_s^3 / r_w^2. With r_s = 18 and r_w = 0.2, length scale is huge; a result in the order of 2 km is reasonable, matching 1944 m.
Why Other Options Are Wrong:
- 2000 m: Rounds too loosely; exact computation gives 1944 m.
- 15000 m: Exaggerates by ~7.7×; inconsistent with volume equality.
- 1855 m: Underestimates the exact ratio.
- None of these: Incorrect since 1944 m is attainable.
Common Pitfalls:
- Forgetting to convert 4 mm to 0.4 cm.
- Using diameter instead of radius in formulas.
- Failing to convert final length from cm to m.
Final Answer:
1944 m
Discussion & Comments