To calculate the cost of copper required to produce 100 meters of wire with a diameter of \(6.304 \times 10^{-3}\) meters, we can follow these steps:
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Calculate the volume of the copper wire. The volume \(V\) of a cylinder (which is the shape of the wire) is given by the formula: \[ V = \pi r^2 h \] where \(r\) is the radius and \(h\) is the height (length in this case).
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The radius \(r\) can be calculated from the diameter: \[ r = \frac{\text{diameter}}{2} = \frac{6.304 \times 10^{-3}}{2} = 3.152 \times 10^{-3} , \text{m} \]
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The height (length) \(h\) is 100 meters.
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Substituting the values, we get: \[ V = \pi (3.152 \times 10^{-3})^2 (100) \] \[ V \approx 3.14159 \times (9.923 \times 10^{-6}) \times (100) \] \[ V \approx 3.14159 \times 9.923 \times 10^{-4} \] \[ V \approx 3.12 \times 10^{-3} , \text{m}^3 \]
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Calculate the mass of the copper. The mass \(m\) can be calculated using the density of copper, which is approximately \(8960 , \text{kg/m}^3\): \[ m = \text{density} \times V \] \[ m = 8960 \times 3.12 \times 10^{-3} \] \[ m \approx 27.96 , \text{kg} \]
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Calculate the cost of the copper. The cost of copper can vary, but as a general estimate, let's assume the price of copper is about $4.30 per kilogram (note that prices can fluctuate). Then the cost \(C\) can be calculated as: \[ C = \text{mass} \times \text{price per kg} \] \[ C = 27.96 \times 4.30 \approx 120.43 \]
Hence, the estimated cost of the copper required to produce 100 meters of wire of the specified diameter is approximately $120.43.
(Note: Please verify the current price of copper, as this value can vary significantly over time and across different markets.)