A copper wire of length 36 m and diameter 2 mm is melted to form a sphere. The radius of the sphere (in cm) is

### Concept & Volume Conservation When a metallic object like a wire (which is cylindrical) is melted and cast into a sphere, the volume remains constant. $$Volume_{cylinder} = \pi r^2 h$$ $$Volume_{sphere} = \frac{4}{3}\pi R^3$$ ### Step-by-Step Solution 1. Identify wire dimensions: Length $h = 36 \text{ m} = 3600 \text{ cm}$. Diameter = $2 \text{ mm} = 0.2 \text{ cm}$, so radius $r = 0.1 \text{ cm}$. 2. Calculate the volume of the wire: $V_{wire} = \pi (0.1)^2 (3600) = \pi (0.01) (3600) = 36\pi \text{ cm}^3$. 3. Let the radius of the newly formed sphere be $R$. 4. Set the volume of the sphere equal to the wire's volume: $\frac{4}{3}\pi R^3 = 36\pi$. 5. Cancel $\pi$ from both sides: $\frac{4}{3}R^3 = 36$. 6. Solve for $R^3$: $R^3 = 36 \times \frac{3}{4} = 9 \times 3 = 27$. 7. Solve for $R$: $R = \sqrt[3]{27} = 3 \text{ cm}$. ### Exam Strategy & Shortcut Ensure all units are converted to centimeters immediately before plugging them into the equation. The equation $\frac{4}{3}R^3 = (0.1)^2 \times 3600$ can be simplified mentally by treating $0.1^2 \times 3600$ as $1 \times 36 = 36$. Then $R^3 = 27$, making $R = 3$. ### Common Pitfall A classic mistake is forgetting to unify the units (meters, millimeters, and centimeters) before beginning the calculation, leading to massive decimal errors. ### Final Answer Therefore, the correct answer is **3**.