QuestionIditional Problems:
1. The loudness level of a heavy snore is 69 dB . The loudness level of a conversation is 60 dB . The loudness level of a whisper is 30 dB . a) How many times as loud as a conversation is a heavy snore? b) How many times as loud as a whisper is a conversation?
2. Most portable music players can produce sounds up to 120 dB . Any sound above 90 dB may cause some hearing loss if the exposure is prolonged. To be safe, experts recommend you keep your MP3 player volume set no higher than $60 \%$ of the maximum. a) Assuming your MP3 player can produce sound as loud as 120 dB , how many times as loud is it at maximum volume than at the recommrended setting? b) How many times as loud is a setting of $75 \%$ of the maximum than $60 \%$ ?

STEP 1

1. The loudness level is measured in decibels (dB).
2. The relationship between loudness levels in dB is logarithmic.
3. The formula to compare loudness levels is $L = 10 \log_{10}(\frac{I}{I_0})$, where $I$ is the intensity and $I_0$ is a reference intensity.
4. We are given specific loudness levels for different sounds and need to find the relative loudness.
_HIGH_LEVEL_APPROACH_ for Problem 1:
1. Calculate the relative loudness of a heavy snore compared to a conversation.
2. Calculate the relative loudness of a conversation compared to a whisper.
_HIGH_LEVEL_APPROACH_ for Problem 2:
1. Calculate the relative loudness of an MP3 player at maximum volume compared to the recommended setting.
2. Calculate the relative loudness of an MP3 player at 75% volume compared to 60% volume.
Problem 1,

STEP 2

STEP 3

Calculate the relative loudness of a heavy snore compared to a conversation.
The formula for comparing loudness levels in decibels is:
$$\text{Difference in dB} = 10 \log_{10} \left( \frac{I_{\text{snore}}}{I_{\text{conversation}}} \right)$$
Given: - Snore = 69 dB - Conversation = 60 dB
$$69 - 60 = 10 \log_{10} \left( \frac{I_{\text{snore}}}{I_{\text{conversation}}} \right)$$
$$9 = 10 \log_{10} \left( \frac{I_{\text{snore}}}{I_{\text{conversation}}} \right)$$
Divide both sides by 10:
$$0.9 = \log_{10} \left( \frac{I_{\text{snore}}}{I_{\text{conversation}}} \right)$$
Convert from logarithmic form to exponential form:
$$\frac{I_{\text{snore}}}{I_{\text{conversation}}} = 10^{0.9}$$
$$\frac{I_{\text{snore}}}{I_{\text{conversation}}} \approx 7.94$$
Problem 1,

STEP 4

Calculate the relative loudness of a conversation compared to a whisper.
Given: - Conversation = 60 dB - Whisper = 30 dB
$$60 - 30 = 10 \log_{10} \left( \frac{I_{\text{conversation}}}{I_{\text{whisper}}} \right)$$
$$30 = 10 \log_{10} \left( \frac{I_{\text{conversation}}}{I_{\text{whisper}}} \right)$$
Divide both sides by 10:
$$3 = \log_{10} \left( \frac{I_{\text{conversation}}}{I_{\text{whisper}}} \right)$$
Convert from logarithmic form to exponential form:
$$\frac{I_{\text{conversation}}}{I_{\text{whisper}}} = 10^{3}$$
$$\frac{I_{\text{conversation}}}{I_{\text{whisper}}} = 1000$$
Problem 2, STEP_1: Calculate the relative loudness of an MP3 player at maximum volume compared to the recommended setting.
Maximum volume = 120 dB Recommended setting = 60% of 120 dB = $0.6 \times 120 = 72$ dB
$$120 - 72 = 10 \log_{10} \left( \frac{I_{\text{max}}}{I_{\text{recommended}}} \right)$$
$$48 = 10 \log_{10} \left( \frac{I_{\text{max}}}{I_{\text{recommended}}} \right)$$
Divide both sides by 10:
$$4.8 = \log_{10} \left( \frac{I_{\text{max}}}{I_{\text{recommended}}} \right)$$
Convert from logarithmic form to exponential form:
$$\frac{I_{\text{max}}}{I_{\text{recommended}}} = 10^{4.8}$$
$$\frac{I_{\text{max}}}{I_{\text{recommended}}} \approx 63095.73$$
Problem 2, STEP_2: Calculate the relative loudness of an MP3 player at 75% volume compared to 60% volume.
75% of 120 dB = $0.75 \times 120 = 90$ dB 60% of 120 dB = 72 dB
$$90 - 72 = 10 \log_{10} \left( \frac{I_{\text{75\%}}}{I_{\text{60\%}}} \right)$$
$$18 = 10 \log_{10} \left( \frac{I_{\text{75\%}}}{I_{\text{60\%}}} \right)$$
Divide both sides by 10:
$$1.8 = \log_{10} \left( \frac{I_{\text{75\%}}}{I_{\text{60\%}}} \right)$$
Convert from logarithmic form to exponential form:
$$\frac{I_{\text{75\%}}}{I_{\text{60\%}}} = 10^{1.8}$$
$$\frac{I_{\text{75\%}}}{I_{\text{60\%}}} \approx 63.10$$