copper rwfinery produces a copper ingot weighing 70 kg. If the copper is drawn into wire whose diameter is 7.50 mm, how nany meters of copper can be obtained frim the ingot? The density of copper is 8.94g/cm^3. Assume the wire is a cylinder.

To determine how many meters of copper wire can be obtained from the ingot, we'll follow these steps:

1. **Calculate the volume of the copper ingot**:
   \[
   \text{Volume of the ingot} = \frac{\text{Mass of the ingot}}{\text{Density of copper}}
   \]
   where:
   - Mass of the ingot \( m = 70 \, \text{kg} \)
   - Density of copper \( \rho = 8.94 \, \text{g/cm}^3 \)

   Convert the mass of the ingot to grams (since the density is in g/cm³):
   \[
   m = 70 \, \text{kg} \times 1000 \, \text{g/kg} = 70000 \, \text{g}
   \]
   Now, calculate the volume:
   \[
   V_{\text{ingot}} = \frac{70000 \, \text{g}}{8.94 \, \text{g/cm}^3} \approx 7828.73 \, \text{cm}^3
   \]

2. **Calculate the volume of the wire**:
   The wire is a cylinder, so its volume \( V_{\text{wire}} \) can be calculated using the formula for the volume of a cylinder:
   \[
   V_{\text{wire}} = \text{cross-sectional area} \times \text{length}
   \]
   The cross-sectional area \( A \) of the wire is:
   \[
   A = \pi \left(\frac{d}{2}\right)^2
   \]
   where \( d = 7.50 \, \text{mm} = 0.75 \, \text{cm} \) is the diameter of the wire.
   \[
   A = \pi \left(\frac{0.75 \, \text{cm}}{2}\right)^2 = \pi \times 0.140625 \, \text{cm}^2 \approx 0.4418 \, \text{cm}^2
   \]

3. **Determine the length of the wire**:
   The volume of the wire \( V_{\text{wire}} \) is equal to the volume of the ingot \( V_{\text{ingot}} \):
   \[
   V_{\text{wire}} = A \times L
   \]
   where \( L \) is the length of the wire in cm. So,
   \[
   L = \frac{V_{\text{ingot}}}{A} = \frac{7828.73 \, \text{cm}^3}{0.4418 \, \text{cm}^2} \approx 17723.86 \, \text{cm}
   \]
   Convert this length to meters:
   \[
   L \approx 177.24 \, \text{meters}
   \]

Thus, approximately **177.24 meters** of copper wire can be obtained from the ingot.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How would the length of the wire change if the diameter was doubled?
2. What would happen to the length if the density of the material was different?
3. How would the length of the wire change if the mass of the ingot increased by 10 kg?
4. What is the effect of temperature on the length of the copper wire?
5. How does the electrical resistance of the wire change with the wire's length?

**Tip:** Remember, the volume of a material remains constant during the transformation of shape (assuming no material loss).

Learn how to calculate the length of copper wire that can be obtained from a 70 kg ingot using geometric formulas. This problem involves calculating the volume of a cylinder and determining wire length. Suitable for high school students studying geometry and dimensional analysis.

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