Understanding Sphere and Wire Volume Conversion

This problem involves the concept of volume conservation. When a solid object like a sphere is melted and recast into another shape, like a wire (which is essentially a long cylinder), its volume remains the same. We need to calculate the volume of the original sphere and equate it to the volume of the wire to find its length.

Calculating Sphere Dimensions

The question gives the diameter of the sphere as 8 cm.

The radius of the sphere ($r_s$) is half of its diameter:

$r_s = \frac{\text{Diameter}}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm}$

Calculating Wire Dimensions

The wire is cylindrical. Its diameter is given as 3 mm.

First, let's convert the wire's diameter to centimeters to maintain consistent units:

$1 \text{ cm} = 10 \text{ mm}$

Wire Diameter = $3 \text{ mm} = \frac{3}{10} \text{ cm} = 0.3 \text{ cm}$

The radius of the wire ($r_w$) is half of its diameter:

$r_w = \frac{\text{Diameter}}{2} = \frac{0.3 \text{ cm}}{2} = 0.15 \text{ cm}$

Calculating Sphere Volume

The formula for the volume of a sphere ($V_s$) is:

$V_s = \frac{4}{3}\pi r_s^3$

Substituting the radius of the sphere:

$V_s = \frac{4}{3}\pi (4 \text{ cm})^3 = \frac{4}{3}\pi (64) \text{ cm}^3 = \frac{256}{3}\pi \text{ cm}^3$

Calculating Wire Volume

The wire is treated as a cylinder. The formula for the volume of a cylinder ($V_w$) is:

$V_w = \pi r_w^2 h$

where $h$ is the length of the wire (height of the cylinder).

Substituting the radius of the wire:

$V_w = \pi (0.15 \text{ cm})^2 h = \pi (0.0225) h \text{ cm}^3$

Equating Volumes to Find Wire Length

Since the volume of the metal remains constant:

$V_s = V_w$

$\frac{256}{3}\pi \text{ cm}^3 = \pi (0.0225) h \text{ cm}^3$

Now, we solve for $h$:

$h = \frac{\frac{256}{3}\pi}{0.0225\pi} \text{ cm}$

Cancel out $\pi$:

$h = \frac{256}{3 \times 0.0225} \text{ cm} = \frac{256}{0.0675} \text{ cm}$

$h \approx 3792.59 \text{ cm}$

Converting Length to Meters

The options are given in meters, so we need to convert the calculated length from centimeters to meters.

$1 \text{ m} = 100 \text{ cm}$

$h = \frac{3792.59}{100} \text{ m} \approx 37.9259 \text{ m}$

Rounding to one decimal place, the length of the wire is approximately 37.9 m.