A solid sphere of radius 6 cm is completely melted and recast into a long, thin wire in the shape of a right circular cylinder with radius 0.2 cm. Assuming no loss of material during melting and recasting, what is the length of the wire (in metres)?

Introduction / Context:
This problem is about conservation of volume when a solid is melted and recast into another shape. A solid sphere is melted to form a cylindrical wire. Since there is no loss of material, the volume of the sphere must be equal to the volume of the cylinder. By using the standard formulas for the volumes of a sphere and a cylinder, we can equate these volumes and solve for the unknown length of the wire. This is a common type of question in geometry and mensuration sections of aptitude exams.

Given Data / Assumptions:

• Radius of the sphere, R = 6 cm.
• The sphere is melted and recast into a cylindrical wire.
• Radius of the cylindrical wire, r = 0.2 cm.
• Length of the wire = L cm (to be found).
• No loss of material during melting and recasting, so volumes are equal.

Concept / Approach:
The volume of a sphere is given by (4 / 3) * π * R^3. The volume of a right circular cylinder is π * r^2 * L. Because the material is conserved, the volume of the original sphere must equal the volume of the resulting cylinder. We set (4 / 3) * π * R^3 = π * r^2 * L, cancel π, and solve for L. After finding L in centimetres, we convert the result to metres as required by the question.

Step-by-Step Solution:
Volume of the sphere = (4 / 3) * π * R^3.Here R = 6 cm, so R^3 = 6^3 = 216.Thus volume of sphere = (4 / 3) * π * 216 = 4 * 72 * π = 288π cubic centimetres.Volume of the cylindrical wire = π * r^2 * L.Radius of cylinder r = 0.2 cm, so r^2 = (0.2)^2 = 0.04.Thus volume of cylinder = π * 0.04 * L.Equate volumes: 288π = π * 0.04 * L.Cancel π from both sides: 288 = 0.04 * L.Solve for L: L = 288 / 0.04.Compute 288 / 0.04 = 288 * (1 / 0.04) = 288 * 25 = 7200 cm.Convert to metres: 7200 cm = 7200 / 100 m = 72 m.

Verification / Alternative check:
We can quickly check if the numbers are reasonable. The radius of the wire is very small compared to the radius of the sphere, so the wire must be quite long. A length of 72 m (7200 cm) is much larger than the original diameter of the sphere (12 cm), which is intuitive. If we incorrectly used r = 2 cm instead of 0.2 cm, we would get a much smaller length, which would not match the thin-wire description. Our calculations with r = 0.2 cm and 288 / 0.04 = 7200 are consistent and error free.

Why Other Options Are Wrong:
The options 81 m, 80 m, 75 m, and 90 m do not satisfy the exact volume equality condition when used in the formula π * 0.04 * L. Substituting any of these values for L into the cylinder volume formula gives volumes that are not equal to 288π cubic centimetres. Only L = 72 m gives the correct cylindrical volume of 288π cubic centimetres, matching the sphere's volume.

Common Pitfalls:
Students often forget to convert the final answer from centimetres to metres or mistakenly use the diameter instead of the radius in the cylinder volume formula. Another common error is squaring 0.2 incorrectly or mishandling the division by 0.04. Some may also forget to cancel π and carry it through unnecessarily, complicating the arithmetic. Careful unit handling and correct use of the volume formulas ensure a straightforward solution.

Final Answer:
The length of the wire formed by melting the sphere is 72 m.