A solid metal sphere of radius 6 cm is melted and recast into a long cylindrical wire of radius 0.2 cm. Assuming no loss of material, what is the length of the wire (in metres)?

Introduction:
This question combines solid geometry and volume conservation. A sphere is melted and reshaped into a cylindrical wire. Since no material is lost, the volume of the sphere must equal the volume of the cylinder. From this equality we can compute the unknown length of the wire.

Given Data / Assumptions:

• Radius of the sphere, R = 6 cm.
• Radius of the cylindrical wire, r = 0.2 cm.
• No loss of material during melting and recasting.
• We need to find the length L of the wire, and express it in metres.

Concept / Approach:
Use standard volume formulas:
Volume of sphere = (4/3) * π * R³.Volume of cylinder = π * r² * L.Since the sphere is completely converted into the cylinder, we set these two volumes equal and solve for L. Finally we convert the length from centimetres to metres.

Step-by-Step Solution:
Compute volume of the sphere:Volume_sphere = (4/3) * π * 6³ = (4/3) * π * 216 = 288π cm³.Volume of the cylinder:Volume_cylinder = π * r² * L = π * (0.2)² * L = π * 0.04 * L.Set volumes equal:288π = π * 0.04 * L.Cancel π on both sides:288 = 0.04 * L.Solve for L:L = 288 / 0.04.0.04 = 4/100, so L = 288 * (100/4) = 288 * 25 = 7200 cm.Convert to metres: 7200 cm = 72 m.

Verification / Alternative check:
We can quickly check: radius of cylinder is very small (0.2 cm), so we expect a long wire. A length of 72 m = 7200 cm is reasonable when compared with the volume 288π. If we plug L = 72 m back in (as 7200 cm), the cylinder volume becomes π * 0.04 * 7200 = π * 288 = 288π cm³, which matches the original sphere volume exactly.

Why Other Options Are Wrong:
Lengths such as 81 m, 80 m, 75 m or 70 m would give cylinder volumes that either exceed or fall short of 288π cm³. Only 72 m keeps the volume exactly equal, which is necessary because no metal is lost in the process.

Common Pitfalls:
Typical mistakes include forgetting to cube the radius for the sphere, squaring the radius incorrectly for the cylinder, or not converting centimetres to metres at the end. Some learners also forget to cancel π from both sides, which complicates the algebra unnecessarily.

Final Answer:
The length of the cylindrical wire is 72 m.