To solve the problem, we can use the following concepts:

1. Power Dissipated by a Resistor:
   The power $P$ dissipated by a resistor is given by: $P = \frac{V^2}{R}$ where $V$ is the potential difference (voltage) and $R$ is the resistance.

2. Resistance of a Wire:
   The resistance $R$ of a wire is related to its resistivity $\rho$, length $L$, and cross-sectional area $A$ by: $R = \rho \frac{L}{A}$

Part (a): Finding the Length of the Wire for 77.0 V

We are given:

• Power, $P = 7490 \, \text{W}$
• Voltage, $V = 77.0 \, \text{V}$
• Cross-sectional area, $A = 3.30 \times 10^{-6} \, \text{m}^2$
• Resistivity of Nichrome, $\rho = 5.00 \times 10^{-7} \, \Omega \cdot \text{m}$

First, solve for the resistance using the power formula: $P = \frac{V^2}{R}$ Rearrange to find $R$: $R = \frac{V^2}{P} = \frac{(77.0 \, \text{V})^2}{7490 \, \text{W}} = \frac{5929 \, \text{V}^2}{7490 \, \text{W}} = 0.7915 \, \Omega$

Next, use the resistance formula to solve for the length $L$: $R = \rho \frac{L}{A}$ Rearrange to find $L$: $L = \frac{R A}{\rho} = \frac{(0.7915 \, \Omega)(3.30 \times 10^{-6} \, \text{m}^2)}{5.00 \times 10^{-7} \, \Omega \cdot \text{m}}$ $L = \frac{2.61195 \times 10^{-6} \, \Omega \cdot \text{m}^2}{5.00 \times 10^{-7} \, \Omega \cdot \text{m}} = 5.224 \, \text{m}$

So, the length of the wire is approximately 5.22 meters.

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Part (b): Finding the Length of the Wire for 114 V

For the second part, we want the same power dissipation (7490 W), but now with a voltage of 114 V. Using the same power formula: $R = \frac{V^2}{P} = \frac{(114 \, \text{V})^2}{7490 \, \text{W}} = \frac{12996 \, \text{V}^2}{7490 \, \text{W}} = 1.735 \, \Omega$

Now solve for the new length $L$: $L = \frac{R A}{\rho} = \frac{(1.735 \, \Omega)(3.30 \times 10^{-6} \, \text{m}^2)}{5.00 \times 10^{-7} \, \Omega \cdot \text{m}}$ $L = \frac{5.7255 \times 10^{-6} \, \Omega \cdot \text{m}^2}{5.00 \times 10^{-7} \, \Omega \cdot \text{m}} = 11.451 \, \text{m}$

So, the length of the wire should be approximately 11.45 meters to achieve the same power dissipation with 114 V.

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Summary of Results:

• (a) The length of the wire for 77.0 V is approximately 5.22 m.
• (b) The length of the wire for 114 V is approximately 11.45 m.

Do you have any questions or want more details on any step?

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Here are 5 follow-up questions you might find interesting:

1. How does the length of the wire affect its resistance?
2. What happens to the power dissipation if the cross-sectional area changes?
3. How would the dissipation rate change if the material's resistivity were different?
4. What would be the length if the power dissipation was reduced to 5000 W at 77.0 V?
5. How does temperature affect the resistivity of Nichrome wire?

Tip: Resistance increases linearly with the length of a wire, meaning doubling the length will double the resistance.