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Chapter 1: Problem 72

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

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

Step by step solution

01

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.
02

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.
03

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.

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