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Convert the following decibels to numbers:

(a) -140 dB
(b) -3 dB
(c) 20 dB
(d) 42 dB

1 Answer

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Final answer:

The decibel levels provide a logarithmic scale for measuring sound intensity, with -140 dB being much quieter than the threshold of hearing, -3 dB slightly less intense, 20 dB indicating a sound 100 times more intense, and 42 dB indicating a sound 15,848 times more intense compared to the reference level which is the threshold of human hearing.

Step-by-step explanation:

The question concerns converting decibel levels to ratios of power intensities or comparing the intensity of sounds in decibels. Decibels (dB) are a logarithmic unit that indicates the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level. Given that the decibel equation for sound intensity level β in decibels is defined as β(dB) = 10 ⋅ log10(I/I0), where I is the intensity of the sound and I0 is the reference sound intensity, usually taken to be the threshold of human hearing (1x10-12 W/m2).

To address the student's examples:

  • -140 dB: This level would indicate a sound intensity ratio much lower than the reference threshold of hearing.
  • -3 dB: A sound that is slightly less intense than the reference level.
  • 20 dB: This level signifies a sound that is 100 times more intense than the reference level.
  • 42 dB: A sound at this level is significantly more intense than the reference level, denoting a sound about 15,848 times more powerful.

Also, if a sound is twice as intense as a 90 dB sound, its level would be slightly more than 3 dB higher, because a doubling in sound intensity corresponds to an increase of approximately 3 dB. Conversely, a sound that is one-fifth as intense as a 90 dB sound would be 7 dB less, as each 10-fold decrease in intensity corresponds to a 10 dB drop.

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