Question

There are 16 similar machines located radially and equally distanced from a fixed sound receiver. While operating, each machine records 60 dB sound level at the receiver. Assuming 70 dB to be the highest sound level allowed as per the industrial sound pollution norms, the total number of machines allowed to operate simultaneously without violating the norms is _________. (rounded off to the nearest integer)

Hint:

When multiple sound sources are involved, the total sound level increases logarithmically. Use this property to calculate the number of sources allowed under a given noise limit.

Answer 1

Given:
Each machine's sound level is 60 dB.
The maximum permissible sound level is 70 dB.
There are 16 machines positioned radially and at equal distances from the sound receiver.
Sound level in decibels (dB) is measured on a logarithmic scale. The combined sound level from multiple sources is calculated using the formula:
\[L_{{total}} = 10 \times \log_{10} \left( \sum_{i=1}^n 10^{L_i/10} \right)\]Where:
\( L_i \) denotes the sound level from an individual source.
\( n \) represents the total number of sources.
Step 1: With each machine emitting 60 dB, the total sound level from \( n \) machines is:
\[L_{{total}} = 10 \times \log_{10} \left( n \times 10^{60/10} \right) = 10 \times \log_{10} \left( n \times 10^6 \right)\]This simplifies to:
\[L_{{total}} = 10 \times \log_{10} (n) + 10 \times \log_{10} (10^6)\]And further to:
\[L_{{total}} = 10 \times \log_{10} (n) + 60\]Step 2: To ensure the total sound level does not exceed 70 dB, the following inequality must hold:
\[10 \times \log_{10} (n) + 60 \leq 70\]Subtracting 60 from both sides yields:
\[10 \times \log_{10} (n) \leq 10\]Dividing by 10 gives:
\[\log_{10} (n) \leq 1\]Exponentiating both sides with base 10:
\[n \leq 10^1 = 10\]Therefore, a maximum of 10 machines are permitted to operate concurrently.