Sound pressure levels were measured at a location over four consecutive 15-minute intervals as 65 dB, 70 dB, 68 dB, and 72 dB. The equivalent continuous sound level (L_eq) over the total one-hour period is ______ dB (rounded off to two decimal places).

Question

Which one of the following gases is primarily responsible for the formation of photochemical smog?

Answer 1

Decibels cannot be averaged arithmetically because the decibel scale is logarithmic, representing sound intensity (a linear, energy-like quantity) compressed by a factor of 10 log10; averaging dB values directly would understate the true energy content of the loudest intervals. The correct method first "undoes" the log scale for each reading, converting 65, 70, 68, and 72 dB back into their proportional linear intensities using $I_i \propto 10^{L_i/10}$, giving relative magnitudes of roughly 3.16, 10.0, 6.31, and 15.85 (all in units of $10^6$). Averaging these four linear intensities directly (since all four intervals are equal length, this is a simple arithmetic mean) gives about $8.83 \times 10^6$. Converting this averaged linear intensity back into a decibel value using $L_{eq}=10\log_{10}(\text{average linear intensity})$ restores the answer to the logarithmic scale, giving approximately 69.43 dB. Notice this result sits closer to the two loudest readings (70 and 72 dB) than a naive arithmetic average of the four dB values (which would give exactly 68.75 dB) - this is expected, because energy averaging always weights the louder intervals more heavily than a straight numerical average would.

\[\boxed{L_{eq} \approx 69.43\ \text{dB}}\]

Answer 2

Decibels cannot be averaged arithmetically because the decibel scale is logarithmic, representing sound intensity (a linear, energy-like quantity) compressed by a factor of 10 log10; averaging dB values directly would understate the true energy content of the loudest intervals. The correct method first "undoes" the log scale for each reading, converting 65, 70, 68, and 72 dB back into their proportional linear intensities using $I_i \propto 10^{L_i/10}$, giving relative magnitudes of roughly 3.16, 10.0, 6.31, and 15.85 (all in units of $10^6$). Averaging these four linear intensities directly (since all four intervals are equal length, this is a simple arithmetic mean) gives about $8.83 \times 10^6$. Converting this averaged linear intensity back into a decibel value using $L_{eq}=10\log_{10}(\text{average linear intensity})$ restores the answer to the logarithmic scale, giving approximately 69.43 dB. Notice this result sits closer to the two loudest readings (70 and 72 dB) than a naive arithmetic average of the four dB values (which would give exactly 68.75 dB) - this is expected, because energy averaging always weights the louder intervals more heavily than a straight numerical average would.

\[\boxed{L_{eq} \approx 69.43\ \text{dB}}\]

Sound pressure levels were measured at a location over four consecutive 15-minute intervals as 65 dB, 70 dB, 68 dB, and 72 dB. The equivalent continuous sound level (L_eq) over the total one-hour period is ______ dB (rounded off to two decimal places).

Show Hint

Correct Answer: 69.43

Solution and Explanation

Questions Asked in GATE ES exam

Sound pressure levels were measured at a location over four consecutive 15-minute intervals as 65 dB, 70 dB, 68 dB, and 72 dB. The equivalent continuous sound level (Leq) over the total one-hour period is ______ dB (rounded off to two decimal places).

Show Hint

Correct Answer: 69.43

Solution and Explanation

Questions Asked in GATE ES exam

Sound pressure levels were measured at a location over four consecutive 15-minute intervals as 65 dB, 70 dB, 68 dB, and 72 dB. The equivalent continuous sound level (L_eq) over the total one-hour period is ______ dB (rounded off to two decimal places).