Part 1: Power loss due to cable's resistance

The machine requires 10 kW of electrical power, and it operates at 100 V. First, we can calculate the current $I$ flowing through the cable using the formula for power:

$P = V \times I$

Where:

• $P$ is the power (10 kW = 10,000 W),
• $V$ is the voltage (100 V),
• $I$ is the current.

Rearranging the formula to find $I$:

$I = \frac{P}{V} = \frac{10,000}{100} = 100 \, \text{A}$

Now, the power lost due to the resistance in the cable can be found using the formula for power loss in a resistor:

$P_{\text{loss}} = I^2 \times R$

Where:

• $P_{\text{loss}}$ is the power lost in the cable,
• $I$ is the current (100 A),
• $R$ is the resistance of the cable (100 Ω).

Substituting the values:

$P_{\text{loss}} = (100)^2 \times 100 = 10,000 \times 100 = 1,000,000 \, \text{W} = 1,000 \, \text{kW}$

Thus, the power lost to the cable’s resistance is 1,000 kW.

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Part 2: Transformer turns ratio

To reduce the power loss by a factor of 104, the voltage supplied to the cable must be increased (while reducing the current). A transformer is used for this purpose. The power loss is proportional to $I^2$, so to reduce the power loss by a factor of 104, the current must be reduced by a factor of:

$\sqrt{104} \approx 10.2$

The transformer increases the voltage and reduces the current proportionally. The turns ratio $n$ of the transformer (primary to secondary) is related to the voltage and current as:

$n = \frac{V_{\text{secondary}}}{V_{\text{primary}}} = \frac{I_{\text{primary}}}{I_{\text{secondary}}}$

Since we want the current reduced by a factor of 10.2, the turns ratio of the transformer must also be 10.2:1.

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Summary:

• The power lost to the cable’s resistance is 1,000 kW.
• The required turns ratio for the transformer is 10.2:1.

Would you like a further breakdown of these steps or more detailed explanations?

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Here are 5 follow-up questions for further exploration:

1. How is power loss affected if the resistance of the cable is halved?
2. What would the power loss be without a transformer if the operating voltage was increased to 200 V?
3. How does increasing the transformer’s turns ratio affect the machine's operation?
4. If the transformer is 95% efficient, how does this affect the overall system power loss?
5. Can you explain how transformer efficiency impacts the power delivered to the machine?

Tip: Transformers allow voltage transformation to reduce current and minimize power loss in transmission systems, which is why they are essential in long-distance power delivery.