Understanding the Problem
To find the length of a copper wire with a specified weight and diameter, we will use the relationships between mass, volume, and density.
Given Data
- Mass of copper wire: 13.2 kg
- Diameter of wire: 4 mm
- Density of copper: 8.4 g/cm³
Step 1: Convert Units
- Convert mass from kg to grams:
13.2 kg = 13200 g
- Convert diameter to radius in centimeters:
Diameter = 4 mm = 0.4 cm
Radius = Diameter / 2 = 0.2 cm
Step 2: Calculate Volume of Copper
Using the formula for density (Density = Mass/Volume), we can rearrange this to find the volume:
Volume = Mass / Density
- Volume of copper wire = 13200 g / 8.4 g/cm³
Volume = 1571.43 cm³
Step 3: Calculate the Volume of the Wire
The volume of a cylindrical wire can be calculated using:
Volume = π × radius² × length
- Rearranging gives:
Length = Volume / (π × radius²)
- Plugging in the values:
Length = 1571.43 cm³ / (π × (0.2 cm)²)
Length = 1571.43 cm³ / (π × 0.04 cm²)
Step 4: Calculate Final Length
- Length ≈ 12500 cm
Conclusion
The length of the 13.2 kg copper wire with a diameter of 4 mm is approximately 12500 cm, or 125 meters. This demonstrates how to connect mass, volume, and geometry in practical applications.