A solid copper sphere of radius 9 cm is melted and recast into a long wire of radius 1 mm. What is the length of the wire, in metres? (Take π = 22/7)

Introduction / Context:
This question applies the principle of volume conservation when a solid is melted and recast into another shape. The copper sphere and the resulting copper wire have the same total volume. To find the length of the wire, we equate the volume of the original sphere to the volume of the cylindrical wire and solve for the unknown length. This is a standard type of mensuration problem in competitive exams.

Given Data / Assumptions:

• Radius of the sphere r_s = 9 cm.
• Radius of the wire r_w = 1 mm = 0.1 cm.
• Volume of the sphere = 4/3 * π * r_s^3.
• Volume of the wire (cylinder) = π * r_w^2 * L, where L is the length in cm.
• π is taken as 22/7 and copper volume is conserved during melting and recasting.

Concept / Approach:
Since the material is the same and nothing is lost, the volume of the original sphere equals the volume of the resulting cylindrical wire. We first write an expression for the sphere volume, then an expression for the cylinder volume. Setting these equal allows us to solve for L, the length of the wire in centimetres. Finally, we convert the length from centimetres to metres by dividing by 100.

Step-by-Step Solution:
Volume of sphere = (4 / 3) * π * r_s^3 = (4 / 3) * π * 9^3.Compute 9^3 = 729, so volume of sphere = (4 / 3) * π * 729 = 972 * π cubic centimetres.Volume of wire (cylinder) = π * r_w^2 * L = π * (0.1)^2 * L = π * 0.01 * L.Equate volumes: 972 * π = π * 0.01 * L, so 972 = 0.01 * L.Hence, L = 972 / 0.01 = 97200 cm, which is 97200 / 100 = 972 metres.

Verification / Alternative check:
To cross check, observe that the radius ratio between sphere and wire is 9 cm to 0.1 cm, which is 90 to 1. Since volume scales with the square of radius times length for a cylinder, the length must be large. A quick sense check confirms that 972 metres is a reasonable value for such a thin wire recast from a sphere of radius 9 cm. Any value much smaller would indicate that the radius conversion or unit conversion was mishandled.

Why Other Options Are Wrong:
Option 1166.4 metres and 1458 metres come from incorrect algebra or misapplied formulas, possibly mixing radius and diameter or using inconsistent units. Option 777.6 metres underestimates the required length and does not satisfy the volume equality when substituted back. Option 900 metres is a rounded guess without basis in the exact calculation and does not keep the volumes equal.

Common Pitfalls:
The most common mistakes are forgetting to convert 1 mm into 0.1 cm, using the diameter instead of the radius in cylinder volume, or failing to square the radius. Some students also forget to convert from centimetres to metres at the end or incorrectly cancel π before setting up the main equation, leading to algebra errors.

Final Answer:
The length of the wire formed by melting the sphere is 972 metres.