Question: (a) What is the decibel level of a sound that is twice as intense as a \(90.0\;{\rm{dB}}\)sound? (b) What is the decibel level of a sound that is one-fifth as intense as a \(90.0\;{\rm{dB}}\)?

The intensity level of the first sound is \(90.0\;{\rm{dB}}\)

The intensity level of the other sounds are twice and one-fifth as intense as the given one.

Decibel is the unit of sound intensity. It is given as ten times the logarithm of the intensity ratios with the threshold.

Now for the double intensity, the intensity level is,

\(\begin{align}d &= 10\log \frac{{2 \times {{10}^{ - 3}}}}{{{{10}^{ - 12}}}}\\ &= 10\log \left( {2 \times {{10}^9}} \right)\\ &= 93\;{\rm{dB}}\end{align}\)

Therefore, the intensity level is \(93\;{\rm{dB}}\).

Now for the one-fifth intensity, the intensity level is,

\(\begin{align}d &= 10\log \frac{{\frac{1}{5} \times {{10}^{ - 3}}}}{{{{10}^{ - 12}}}}\\ &= 10\log \left( {0.2 \times {{10}^9}} \right)\\ &= 83\;{\rm{dB}}\end{align}\)

Therefore, the intensity level is \(83\;{\rm{dB}}\).