Calculating Wire Diameter from Melted Sphere

The problem involves finding the diameter of a wire formed by melting a copper sphere. This requires applying the principle of volume conservation.

Step 1: Identify Given Information and Formulas

• Sphere diameter = 12 cm. Therefore, sphere radius $R = \frac{12}{2} = 6$ cm.
• Wire length $L = 48$ cm.
• Let the wire's radius be $r$ and its diameter be $d$.
• Volume of a sphere: $V_{sphere} = \frac{4}{3}\pi R^3$.
• Volume of a cylinder (wire): $V_{wire} = \pi r^2 L$.

Step 2: Calculate the Volume of the Copper Sphere

Using the formula for the volume of a sphere:

$V_{sphere} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (6 \text{ cm})^3$

$V_{sphere} = \frac{4}{3}\pi (216 \text{ cm}^3) = 4 \times 72 \pi \text{ cm}^3 = 288\pi \text{ cm}^3$.

Step 3: Calculate the Radius of the Wire

Since the sphere is melted and reformed into a wire, the volume remains constant ($V_{sphere} = V_{wire}$).

$V_{wire} = \pi r^2 L = 288\pi \text{ cm}^3$.

Substitute the known length of the wire ($L = 48$ cm):

$\pi r^2 (48 \text{ cm}) = 288\pi \text{ cm}^3$.

Divide both sides by $48\pi \text{ cm}$ to find $r^2$:

$r^2 = \frac{288\pi \text{ cm}^3}{48\pi \text{ cm}} = 6 \text{ cm}^2$.

Solve for $r$:

$r = \sqrt{6} \text{ cm}$.

Step 4: Calculate the Diameter of the Wire

The diameter $d$ is twice the radius $r$:

$d = 2r = 2 \times \sqrt{6} \text{ cm} = 2\sqrt{6} \text{ cm}$.

Conclusion

The diameter of the wire is $2\sqrt{6}$ cm.