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 \(2077.44 \textrm{ cm}\).

Step by step solution

01

Calculate the volume of the copper wire

To find the volume of the copper wire, we can use the formula: Volume \(= \frac{\text{mass}}{\text{density}}\) We are given the mass of the wire as 22 g and the density of copper as \(8.96 \textrm{ g/cm}^3\). Plugging in the values, we get: Volume \(= \frac{22 \textrm{ g}}{8.96 \, \textrm{g/cm}^3}\)
02

Calculate the cross-sectional area of the wire

The copper wire has a circular cross-section, and we are given its diameter as 0.25 mm. To calculate the cross-sectional area, we first need to find the wire's radius. The radius of the wire is half of the diameter: Radius \(= \frac{\text{diameter}}{2} = \frac{0.25 \textrm{mm}}{2} = 0.125 \textrm{mm}\) Now, to find the cross-sectional area of the wire, we use the formula for the area of a circle: Area \(= \pi (\text{radius})^2\) Plugging in the value of the radius, we get: Area \(= \pi (0.125 \textrm{mm})^2\) It is important to convert the radius to cm before proceeding: Radius \(= 0.0125 \textrm{ cm}\) Area \(= \pi (0.0125 \textrm{cm})^2\)
03

Calculate the length of the wire

Now that we have the volume and cross-sectional area of the wire, we can find its length using the formula: Length \(= \frac{\text{volume}}{\text{area}}\) Plugging in the values, we get: Length \(= \frac{\frac{22}{8.96}}{\pi (0.0125)^2}\) Now, solve for the length: Length \(= \frac{22}{8.96 \times \pi (0.0125)^2}\textrm{ cm}\) Length \(= 2077.44 \textrm{ cm}\) So, the length of the copper wire is approximately 2077.44 cm.

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