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:
The sound level of each machine = 60 dB
The highest sound level allowed = 70 dB
There are 16 machines located radially and equally distanced from the sound receiver
The sound level in decibels (dB) is a logarithmic scale, and the total sound level from multiple sources is given by: \[ L_{{total}} = 10 \times \log_{10} \left( \sum_{i=1}^n 10^{L_i/10} \right) \] Where:
\( L_i \) is the sound level from each source
\( n \) is the number of sources
Step 1: The sound level from each machine is 60 dB. The total sound level with \( n \) machines is: \[ L_{{total}} = 10 \times \log_{10} \left( n \times 10^{60/10} \right) = 10 \times \log_{10} \left( n \times 10^6 \right) \] \[ L_{{total}} = 10 \times \log_{10} (n) + 10 \times \log_{10} (10^6) \] \[ L_{{total}} = 10 \times \log_{10} (n) + 60 \] Step 2: The total sound level must not exceed 70 dB, so: \[ 10 \times \log_{10} (n) + 60 \leq 70 \] \[ 10 \times \log_{10} (n) \leq 10 \] \[ \log_{10} (n) \leq 1 \] \[ n \leq 10^1 = 10 \] Thus, the total number of machines allowed to operate simultaneously is 10.