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Let's solve the given problem using the formula provided:

\[
L_2 = L_1 + 20 \log \left( \frac{r_1}{r_2} \right)
\]

where:
- \( L_1 \) is the initial loudness at distance \( r_1 \).
- \( L_2 \) is the loudness at a different distance \( r_2 \).
- \( r_1 \) is the initial distance.
- \( r_2 \) is the new distance.

### Given Data:
- \( r_1 = 10 \) meters
- \( L_1 = 84 \) dB
- \( r_2 = 30 \) meters

#### Part (a): Find \( L_2 \) at 30 meters.
\[
L_2 = 84 + 20 \log \left( \frac{10}{30} \right)
\]
\[
L_2 = 84 + 20 \log \left( \frac{1}{3} \right)
\]
\[
L_2 = 84 + 20 \times (-0.4771)
\]
\[
L_2 = 84 - 9.542
\]
\[
L_2 \approx 74.46 \, \text{dB}
\]

So, the sound level at 30 meters from the speaker is approximately \( 74.46 \, \text{dB} \).

#### Part (b): Find the distance \( r_2 \) for the volume to be halved.
A halving of the volume in decibels means a reduction of 3 dB, so \( L_2 = 84 - 3 = 81 \) dB.

\[
81 = 84 + 20 \log \left( \frac{10}{r_2} \right)
\]
\[
81 - 84 = 20 \log \left( \frac{10}{r_2} \right)
\]
\[
-3 = 20 \log \left( \frac{10}{r_2} \right)
\]
\[
\log \left( \frac{10}{r_2} \right) = \frac{-3}{20}
\]
\[
\log \left( \frac{10}{r_2} \right) = -0.15
\]
\[
\frac{10}{r_2} = 10^{-0.15}
\]
\[
\frac{10}{r_2} \approx 0.7079
\]
\[
r_2 \approx \frac{10}{0.7079} \approx 14.12 \, \text{meters}
\]

So, you have to move approximately 14.12 meters away from the speaker for the volume to be halved.

### Do you want more details or have any other questions?

### Related Questions:
1. What would be the sound level at 50 meters from the speaker?
2. If the initial loudness was 90 dB at 5 meters, what would be the loudness at 20 meters?
3. How does the sound level change if you move from 10 meters to 5 meters?
4. What distance is required to reduce the loudness by 10 dB?
5. How does the formula change if we consider sound intensity instead of loudness?

**Tip:** Remember, sound intensity decreases with the square of the distance, while sound level in dB decreases logarithmically.

10 meters from a speaker, the volume is 84 dB. a) What is the sound level 30 meters from the speaker? b) How far from the speaker do you have to move for the volume to be halved?

Learn how to calculate the sound level at different distances from a speaker using the formula L2 = L1 + 20 log(r1/r2). Given the initial loudness is 84 dB at 10 meters, find the sound level at 30 meters and the distance for halving the volume. Suitable for Grades 10-12.

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