Given: 
a. 20% Volume = 700Hz(-54dB)
b. 50% Volume = 937Hz(-41dB)
c. 100% Volume = 1044Hz(-29dB)

1. Calculate each sound's source wavelength using the recorded frequencies. Simplify the answer.
2. Assume that the signals have 1.98 volts RMS(root mean square) and are being corrupted by 30 mV RMS noise. Compute the S/N ratio.

### 1. **Calculate Each Sound's Source Wavelength**

The formula to calculate the wavelength \( \lambda \) of a sound wave is:

\[
\lambda = \frac{v}{f}
\]

Where:
- \( v \) is the speed of sound (approximately 343 m/s at room temperature),
- \( f \) is the frequency in Hz.

Now, we will calculate the wavelength for each of the given frequencies.

#### a. 20% Volume, 700 Hz

\[
\lambda = \frac{343 \, \text{m/s}}{700 \, \text{Hz}} = 0.49 \, \text{m}
\]

#### b. 50% Volume, 937 Hz

\[
\lambda = \frac{343 \, \text{m/s}}{937 \, \text{Hz}} = 0.366 \, \text{m}
\]

#### c. 100% Volume, 1044 Hz

\[
\lambda = \frac{343 \, \text{m/s}}{1044 \, \text{Hz}} = 0.328 \, \text{m}
\]

Thus, the wavelengths for the sounds are:
- At 700 Hz: 0.49 meters
- At 937 Hz: 0.366 meters
- At 1044 Hz: 0.328 meters

### 2. **Compute the S/N Ratio**

The signal-to-noise ratio (S/N ratio) is the ratio of the RMS voltage of the signal to the RMS voltage of the noise, usually expressed in decibels (dB). The formula is:

\[
S/N \, \text{(dB)} = 20 \log_{10}\left(\frac{V_{\text{signal}}}{V_{\text{noise}}}\right)
\]

Where:
- \( V_{\text{signal}} = 1.98 \, \text{V RMS} \),
- \( V_{\text{noise}} = 30 \, \text{mV RMS} = 0.030 \, \text{V RMS} \).

Substituting into the formula:

\[
S/N = 20 \log_{10}\left(\frac{1.98}{0.030}\right)
\]

Now, let’s calculate this.

\[
S/N = 20 \log_{10}(66) \approx 20 \times 1.819 = 36.38 \, \text{dB}
\]

Thus, the signal-to-noise ratio is approximately **36.38 dB**.

---

Would you like further details or have any additional questions?

Here are 5 related questions to explore:
1. How does temperature affect the speed of sound and wavelength calculations?
2. What is the difference between RMS voltage and peak voltage in signal analysis?
3. How does the signal-to-noise ratio influence audio quality in recording systems?
4. Why is it important to reduce noise levels in electrical signals?
5. How do decibels relate to perceived sound intensity by the human ear?

**Tip:** When dealing with frequencies and wavelengths, always ensure to account for environmental factors like temperature, as they can affect the speed of sound significantly.

Given: a. 20% Volume = 700Hz(-54dB), b. 50% Volume = 937Hz(-41dB), c. 100% Volume = 1044Hz(-29dB). 1. Calculate each sound's source wavelength using the recorded frequencies. Simplify the answer. 2. Assume that the signals have 1.98 volts RMS (root mean square) and are being corrupted by 30 mV RMS noise. Compute the S/N ratio.

Learn how to calculate the wavelength of sound at different volumes and frequencies, and compute the signal-to-noise ratio for a signal with 1.98V RMS and 30mV noise. This problem covers wave physics and signal processing concepts with a step-by-step solution.

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