To solve this problem, we need to determine the power level at which a signal must be transmitted such that after traveling 12 km, the signal maintains a minimum Signal to Noise Ratio (SNR) of 10,000:1. Given the background noise of 700 nW and a 10% power loss per kilometer, we can follow these steps:

Step 1: Understanding the signal power at the end of the cable

The power is reduced by 10% per kilometer, which means 90% of the power remains after each kilometer. The signal power at the end of 12 kilometers is given by:

$P_{\text{end}} = P_{\text{transmit}} \times (0.9)^{12}$

where:

• $P_{\text{transmit}}$ is the initial transmitted power,
• 0.9 represents the 90% of power retained after each kilometer,
• 12 is the length of the cable in kilometers.

Step 2: Signal to Noise Ratio (SNR)

The Signal to Noise Ratio is the ratio of the signal power at the end of the cable to the noise power. We're given: $\text{SNR} = \frac{P_{\text{end}}}{P_{\text{noise}}}$ where $P_{\text{noise}} = 700 \text{ nW}$ and the minimum SNR requirement is 10,000:1.

So, the equation for the minimum SNR is: $\frac{P_{\text{end}}}{700 \times 10^{-9}} = 10,000$ $P_{\text{end}} = 700 \times 10^{-9} \times 10,000 = 7 \times 10^{-3} \text{ W} = 7 \text{ mW}$

Thus, the signal power at the end of the 12 km cable must be 7 mW to maintain the required SNR.

Step 3: Transmitted power calculation

We can now calculate the transmitted power needed, given the power loss over 12 km:

$P_{\text{end}} = P_{\text{transmit}} \times (0.9)^{12}$ Substitute $P_{\text{end}} = 7 \text{ mW}$: $7 \times 10^{-3} = P_{\text{transmit}} \times (0.9)^{12}$

Now, calculate $(0.9)^{12}$: $(0.9)^{12} \approx 0.2824$

So: $7 \times 10^{-3} = P_{\text{transmit}} \times 0.2824$ $P_{\text{transmit}} = \frac{7 \times 10^{-3}}{0.2824} \approx 0.0248 \text{ W} = 24.8 \text{ mW}$

Conclusion

The signal must be transmitted at a power level of approximately 24.8 mW to ensure that after 12 km, the signal still has a minimum Signal to Noise Ratio of 10,000:1.

Would you like further clarification or more details on any part of the solution?

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Here are 5 related questions:

1. How does the required transmission power change if the cable length is increased?
2. What if the power loss per kilometer is 15% instead of 10%?
3. How is the SNR affected if the background noise increases?
4. What is the impact on transmission power if a lower SNR of 1,000:1 is acceptable?
5. How could using a cable with lower loss per kilometer improve the overall efficiency?

Tip: In signal transmission, minimizing power loss over long distances can be achieved by using higher-quality cables or signal amplifiers at intervals.