Question

A copper wire (density \(=8.96 \mathrm{~g} / \mathrm{cm}^{3}\) ) has a diameter of \(0.25 \mathrm{~mm}\). If a sample of this copper wire has a mass of \(22 \mathrm{~g}\), how long is the wire?

Short answer

The length of the copper wire is approximately 4998.88 cm or about 49.99 meters.

Step-by-step solution

Find the volume of the wire

We are given the mass as 22 g and the density as 8.96 g/cm³. We can use the formula for density to find the volume: \[density = \frac{mass}{volume}\] From this formula, we can find the volume: \[volume = \frac{mass}{density}\] Substitute the given values into the formula: \[volume = \frac{22}{8.96}\] Now, calculate the volume: \[volume \approx 2.4536 \, \mathrm{cm}^{3}\] So, the volume of the wire is approximately 2.4536 cm³.

Find the cross-sectional area of the wire

We are given the diameter (0.25 mm) of the wire. First, convert the diameter from mm to cm: \[diameter = 0.25 \, \mathrm{mm} \times \frac{1 \, \mathrm{cm}}{10\, \mathrm{mm}} = 0.025 \, \mathrm{cm}\] Now find the radius: \[radius = \frac{diameter}{2} = \frac{0.025}{2} = 0.0125 \, \mathrm{cm}\] The cross-sectional area of the wire is the area of a circle, which can be found using the formula: \[A = \pi r^2\] Substitute the radius value into the formula: \[A = \pi\times (0.0125)^2\] Now, calculate the cross-sectional area: \[A \approx 0.00049087 \, \mathrm{cm}^{2}\] So, the cross-sectional area of the wire is approximately 0.00049087 cm².

Find the length of the wire

Now, we can use the volume and the cross-sectional area to find the length of the wire. Divide the volume of the sample by the cross-sectional area of the wire: \[length = \frac{volume}{cross-sectional\, area}\] Substitute the given values into the formula: \[length = \frac{2.4536}{0.00049087}\] Now, calculate the length: \[length \approx 4998.8824 \, \mathrm{cm}\] So, the length of the wire is approximately 4998.88 cm or about 49.99 meters.