Calculating Sphere Radius from Melted Copper Wire

The problem involves calculating the radius of a sphere formed by melting a copper wire. The key principle is that the volume of the material remains constant during the process of melting and recasting. Therefore, the volume of the original copper wire (a cylinder) is equal to the volume of the resulting sphere.

Steps for Calculation

• Unit Conversion: First, ensure all measurements are in consistent units. We will use centimeters (cm).
  • Wire Length ($L$): $36 \text{ m} = 36 \times 100 \text{ cm} = 3600 \text{ cm}$.
  • Wire Diameter ($d$): $2 \text{ mm} = \frac{2}{10} \text{ cm} = 0.2 \text{ cm}$.
  • Wire Radius ($r_{wire}$): $r_{wire} = \frac{d}{2} = \frac{0.2 \text{ cm}}{2} = 0.1 \text{ cm}$.
• Calculate Volume of Copper Wire: The wire is cylindrical. The formula for the volume of a cylinder is $V_{cylinder} = \pi r^2 h$.
  • $V_{wire} = \pi \times (r_{wire})^2 \times L$
  • $V_{wire} = \pi \times (0.1 \text{ cm})^2 \times (3600 \text{ cm})$
  • $V_{wire} = \pi \times 0.01 \text{ cm}^2 \times 3600 \text{ cm}$
  • $V_{wire} = 36\pi \text{ cm}^3$
• Equate Volumes: The volume of the sphere ($V_{sphere}$) must equal the volume of the wire ($V_{wire}$). The formula for the volume of a sphere is $V_{sphere} = \frac{4}{3} \pi r_{sphere}^3$.
  • $V_{sphere} = V_{wire}$
  • $\frac{4}{3} \pi r_{sphere}^3 = 36\pi \text{ cm}^3$
• Solve for Sphere Radius ($r_{sphere}$):
  • Divide both sides by $\pi$: $\frac{4}{3} r_{sphere}^3 = 36 \text{ cm}^3$.
  • Multiply both sides by $\frac{3}{4}$: $r_{sphere}^3 = 36 \times \frac{3}{4} \text{ cm}^3$.
  • $r_{sphere}^3 = 9 \times 3 \text{ cm}^3 = 27 \text{ cm}^3$.
  • Take the cube root of both sides: $r_{sphere} = \sqrt[3]{27 \text{ cm}^3}$.
  • $r_{sphere} = 3 \text{ cm}$.

Based on the calculations derived directly from the question's parameters, the radius of the sphere is 3 cm.

The provided correct answer is Option D: 2.5 cm.