Question

If transmission and engine system of a tractor produce a noise of 80 dB each, then the overall noise level will be

• A. 80 dB
• B. 160 dB
• C. 83 dB
• D. 113 dB

Hint:

When adding two identical sound levels, the resultant level always increases by exactly 3 dB.
Thus, \(80\text{ dB} + 80\text{ dB} = 83\text{ dB}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Decibels (dB) are logarithmic units used to measure sound power or intensity level.
Because of the logarithmic scale, decibel values cannot be added together using standard arithmetic.

Step 2: Key Formula or Approach:
The total sound pressure level \(L_{\text{total}}\) resulting from multiple independent noise sources is calculated using:
\[ L_{\text{total}} = 10 \log_{10} \left( \sum_{i=1}^{n} 10^{\frac{L_i}{10}} \right) \]

Step 3: Detailed Explanation:
Let the noise levels of the two sources be \(L_1 = 80\text{ dB}\) and \(L_2 = 80\text{ dB}\).
Using the logarithmic addition formula:
\[ L_{\text{total}} = 10 \log_{10} \left( 10^{\frac{80}{10}} + 10^{\frac{80}{10}} \right) \] \[ L_{\text{total}} = 10 \log_{10} \left( 10^8 + 10^8 \right) \] \[ L_{\text{total}} = 10 \log_{10} \left( 2 \times 10^8 \right) \] Applying logarithmic identities:
\[ L_{\text{total}} = 10 \left( \log_{10}(2) + \log_{10}(10^8) \right) \] \[ L_{\text{total}} = 10 \left( 0.3010 + 8 \right) \] \[ L_{\text{total}} = 10 \times 8.3010 = 83.01\text{ dB} \approx 83\text{ dB} \]

Step 4: Final Answer:
The correct option is 3, which corresponds to 83 dB.