Let's address the problem step-by-step.
### Problem Analysis
We need to determine the difference in sound levels between two positions related to an explosion:
1. A person directly below the explosion.
2. A person at a horizontal distance of 220 meters away.
### Sound Level Difference Formula
The difference in sound levels ([tex]\(\Delta \beta\)[/tex]) in decibels (dB) between two points at distances [tex]\(d_1\)[/tex] and [tex]\(d_2\)[/tex] from the sound source can be calculated using the formula:
[tex]\[ \Delta \beta = 20 \cdot \log_{10}\left(\frac{d_2}{d_1}\right) \][/tex]
### Given Values
- [tex]\(L = 220\)[/tex] meters (horizontal distance)
- [tex]\(d_1 = 1\)[/tex] meter (assuming a very close distance directly below the explosion)
- [tex]\(d_2 = 220\)[/tex] meters (the horizontal distance, given)
### Calculation
To find the difference in sound levels using the provided distances:
1. Substitute [tex]\(d_1 = 1\)[/tex] meter and [tex]\(d_2 = 220\)[/tex] meters into the formula:
[tex]\[ \Delta \beta = 20 \cdot \log_{10}(220 / 1) \][/tex]
2. Perform the logarithmic calculation and multiplication:
[tex]\[ \Delta \beta = 20 \cdot \log_{10}(220) \][/tex]
### Solution
The above steps produce the following result:
[tex]\[ \Delta \beta = 46.85 \text{ dB} \][/tex]
### Final Answer
The difference in sound levels is:
[tex]\[
\Delta \beta = 46.85 \text{ dB}
\][/tex]
This expresses how much greater the sound level is for a person directly below the explosion compared to a person 220 meters away.
