Problem 48 - Precalculus with Limits

Question

In Exercises \(47-50,\) use the following information for determining sound intensity. The level of sound \(\beta,\) in decibels, with an intensity of \(I,\) is given by \(\beta=10 \log \left(I I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 47 and \(48,\) find the level of sound \(\boldsymbol{\beta}\) (a) \(I=10^{-11}\) watt per \(\mathrm{m}^{2}\) (rustle of leaves) (b) \(I=10^{2}\) watt per \(\mathrm{m}^{2}\) (jet at 30 meters) (c) \(I=10^{-4}\) watt per \(\mathrm{m}^{2}\) (door slamming) (d) \(I=10^{-2}\) watt per \(\mathrm{m}^{2}\) (siren at 30 meters)

Step-by-Step Solution

Answer

The intensity of sound for (a) rustle of leaves is 10 decibels, (b) jet at 30 meters is 140 decibels, (c) door slamming is 80 decibels, (d) siren at 30 meters is 100 decibels.

Step 1: Finding '\(\beta\)' for 'rustle of leaves'

Replace 'I' with \(10^{-11}\) watt per square meter in formula \(\beta=10 \log \left(I I_{0}\right)\). Calculation: \(\beta = 10 * log(10^{-11}/10^{-12}) = 10* log(10) = 10*1 = 10\) dB. The sound level for the rustle of leaves is 10 decibels.

Step 2: Finding '\(\beta\)' for 'jet at 30 meters'

Replace 'I' with \(10^{2}\) watt per square meter in formula \(\beta=10 \log \left(I I_{0}\right)\). Calculation: \(\beta = 10 * log(10^{2}/10^{-12}) = 10* log(10^{14}) = 10*14 = 140\) dB. The sound level for jet at 30 meters is 140 decibels.

Step 3: Finding '\(\beta\)' for 'door slamming'

Replace 'I' with \(10^{-4}\) watt per square meter in the formula \(\beta=10 \log \left(I I_{0}\right)\). Calculation: \(\beta = 10 * log(10^{-4}/10^{-12}) = 10* log(10^{8}) = 10*8 = 80\) dB. The sound level for door slamming is 80 decibels.

Step 4: Finding '\(\beta\)' for 'siren at 30 meters'

Replace 'I' with \(10^{-2}\) watt per square meter in the formula \(\beta=10 \log \left(I I_{0}\right)\). Calculation: \(\beta = 10 * log(10^{-2}/10^{-12}) = 10* log(10^{10}) = 10*10 = 100\) dB. The sound level for siren at 30 meters is 100 decibels.

Key Concepts

Decibels

The decibel (dB) is a unit used to measure the intensity of a sound or the power level of an electrical signal by comparing it with a given level on a logarithmic scale. In the context of sound, it is a measure of how loud a noise appears to our ears. The faintest sound that a typical human ear can detect is assigned a sound level of 0 dB, while a sound 10 times more powerful is 10 dB, and 100 times more powerful compared to the faintest audible sound is 20 dB.

When calculating sound levels in decibels, we use a reference sound intensity level of \( I_0 = 10^{-12} \text{watt per square meter} \), which is considered the threshold of human hearing. The formula to find the level of sound in decibels for a given intensity \(I\) is \( \beta = 10 \log\left(\frac{I}{I_{0}}\right) \). This logarithmic nature allows us to handle the wide range of sound intensities we can hear in a compact scale.

Logarithmic Scales

Logarithmic scales are indispensable in various fields of science and engineering because they allow us to represent very large or very small numbers in a manageable way. When dealing with quantities that can vary over several orders of magnitude, such as sound intensity, a linear scale becomes impractical. For example, the human ear can hear sounds ranging from \( I_0 \), the faintest sound, up to sounds that are a trillion times more intense.

Logarithms help by expressing those variations in manageable numbers (decibels). The consequence of this scale is that an increase of 10 dB represents a tenfold increase in intensity, which is why decibels are calculated using the logarithm of the ratio of a particular intensity to the reference intensity. The logarithmic nature ensures that each step up the scale represents a doubling of the perceived loudness.

Intensity of Sound

Sound intensity is a measure of the energy of sound waves per unit area. It is a quantitative measure of the 'loudness' or the power carried by sound waves. Sound intensity is usually measured in watts per square meter (\( \text{W/m}^2 \)). It is important to note that intensity is a physical quantity and is distinct from the perceived loudness, which is how humans interpret the intensity of a sound.

The formula \( \beta = 10 \log\left(\frac{I}{I_{0}}\right) \) highlights a direct relationship between the sound intensity and the decibel level. However, due to the human ear's response, the perceived loudness of a sound does not increase linearly with intensity; a sound must have far greater intensity to be perceived as twice as loud. The diverse examples in the exercises - rustle of leaves, a jet engine, a door slamming, and a siren - demonstrate how intensity in watts per square meter corresponds with our logarithmic perception of loudness in decibels.