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Chapter 1: Problem 64

A copper refinery produces a copper ingot weighing \(70 \mathrm{~kg}\). If the copper is drawn into wire whose diameter is \(7.50 \mathrm{~mm}\), how many meters of copper can be obtained from the ingot? The density of copper is \(8.94 \mathrm{~g} / \mathrm{cm}^{3}\). (Assume that the wire is a cylinder whose volume \(V=\pi r^{2} h,\) where \(r\) is its radius and \(h\) is its height or length.)

Short Answer

Expert verified
Approximately 210 meters of copper wire can be obtained from the ingot.

Step by step solution

01

Find the volume of the copper ingot

We are given the mass of the copper ingot and the density of the copper. We can use the following formula to find the volume: Volume = Mass/Density Mass of ingot = 70 kg Since we will later need the density and mass to have matching units, let's first convert mass from kg to g: Mass of ingot = \(70 \times 1000 = 70,000 \mathrm{~g}\) Density of copper = 8.94 g/cm³ Volume of ingot = \(\frac{70,000 \mathrm{~g}}{8.94\mathrm{~g/cm}^{3}}\) Now, calculate the volume of the ingot: Volume of ingot = \(7834.45 \mathrm{~cm}^{3}\)
02

Find the radius of the copper wire

We are given the diameter of the copper wire which is 7.50 mm. To find the radius, we just need to divide the diameter by 2: Radius = Diameter/2 Radius = \(7.50 \mathrm{~mm}/2\) Now, convert the radius from mm to cm as we will later need the radius and volume to have matching units: Radius = \(0.375 \mathrm{~cm}\)
03

Use the volume formula for a cylinder to find the length (height) of wire

The volume formula for a cylinder is: \[V = \pi r^{2} h\] Since we are looking for the height (length) of the wire, rearrange the formula for h: \[h = \frac{V}{\pi r^{2}}\] We've already found the volume of the ingot and the radius of the wire, so we can plug in these values and calculate the height (length) of the wire: \[ h = \frac{7834.45 \mathrm{~cm}^{3}}{\pi (0.375 \mathrm{~cm})^{2}}\] Now, calculate the length (height) of the wire: Length of wire = 21062.25 cm Finally, convert the length from cm to meters: Length of wire = \(210.62 \mathrm{~m}\) So, approximately 210 meters of copper wire can be obtained from the ingot.

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