Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the following information for determining sound intensity. The level of sound \(\beta\) (in decibels) with an intensity \(I\) is $$\beta=10 \log _{10} \frac{I}{I_{0}}$$ 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 49 and \(50,\) find the Ievel of each sound \(\beta\). (a) \(I=10^{-10}\) watt per \(m^{2}\) (quiet room) (b) \(I=10^{-5}\) watt per \(\mathrm{m}^{2}\) (busy street corner) (c) \(I \approx 10^{0}\) watt per \(\mathrm{m}^{2}\) (threshold of pain)

Step-by-Step Solution

Answer

The sound level in a quiet room is 20 decibels; in a busy street corner, it's 70 decibels; and the threshold of pain is around 120 decibels.

Step 1: Apply the formula for the quiet room

The formula given is \( \beta = 10 \log_{10} \frac{I}{I_0} \). Let's apply the values for the quiet room, where \(I = 10^{-10}\): \n\n\( \beta = 10 \log_{10} \frac{10^{-10}}{10^{-12}} \)\n\nUsing the property of logarithms \( \log_{b} \frac{a}{c} = \log_{b} a - \log_{b} c \), we get: \n\n\( \beta = 10 ( \log_{10} 10^{-10} - \log_{10} 10^{-12}) \)

Step 2: Calculate the decibel level for the quiet room

Since \( \log_{b} b^{n} = n \), then \( \beta = 10(-10-(-12)) = 10*2 = 20 \) decibels. So, the sound level in a quiet room is 20 decibels.

Step 3: Apply formula for the busy street corner and calculate

Now let's find for the busy street, where the intensity \(I = 10^{-5}\). \n\nFrom \( \beta = 10 \log_{10} \frac{I}{I_0} \) we get: \n\n \( \beta = 10 \log_{10} \frac{10^{-5}}{10^{-12}} \) which results in \( \beta = 10 ( \log_{10} 10^{-5} - \log_{10} 10^{-12}) \)\n\nUsing the property explained in the first step, finally we have\n\n\( \beta = 10(-5-(-12)) = 10*7 = 70 \) decibels. So, the sound level in a busy street corner is 70 decibels.

Step 4: Apply formula for the threshold of pain and calculate

Finally let's find for the threshold of pain, where the intensity \(I \approx 10^{0}= 1\). \n\nFrom \( \beta = 10 \log_{10} \frac{I}{I_0} \) we get: \n\n \( \beta = 10 \log_{10} \frac{1}{10^{-12}} \) which results in \( \beta = 10 ( \log_{10} 1 - \log_{10} 10^{-12}) \)\n\nRemembering that \( \log_{b} 1 = 0 \), we then find\n\n\( \beta = 10(0-(-12)) = 10*12 = 120 \) decibels. So, the threshold of pain is approximately 120 decibels.

Key Concepts

Logarithmic Equations

When we encounter logarithmic equations, we are dealing with expressions that have the unknown variable within the log function. Solving these equations often involves understanding the definition of a logarithm and employing manipulation strategies based on the properties of logarithms. For instance, the equation given in the sound intensity context, \( \beta = 10 \log_{10} \frac{I}{I_0} \), involves calculating the sound level in decibels which necessitates isolating the log term and then finding the intensity value \( I \).

Consider the underlying principle: if we have \( \log_b(x) = y \), this is equivalent to \( b^y = x \). With this conversion from a logarithmic equation to an exponential one, we can solve for the unknown. Students often find logarithmic equations challenging because they also require familiarity with exponent rules and often, the properties of logarithms which are crucial for simplifying and solving these equations.

Decibel Level Calculations

Decibel level calculations are a measure of sound intensity expressed in decibels (dB). The formula used in these exercises is \( \beta = 10 \log_{10} \frac{I}{I_0} \), where \( \beta \) is the sound level in decibels, \(| I \) is the intensity of the sound in watts per square meter, and \( I_0 \) is the reference sound intensity, considered as the faintest sound the human ear can detect, typically set to \( 10^{-12} \) watts per square meter.

The decibel scale is logarithmic, which reflects how the human ear perceives sound. A tenfold increase in intensity (\( 10 \times \)) results in an increase of 10 decibels in the sound level, while a hundredfold increase in intensity (\( 100 \times \)) would correspond to a 20-decibel increase. This logarithmic nature allows a wide range of sound intensities to be represented in a more compact scale.

Intensity of Sound

Sound intensity is a measure of the power per unit area carried by a sound wave and is measured in watts per square meter (W/m\( ^2 \)). It is a critical concept in understanding how loud a sound is. Intensity is not directly proportional to the loudness sensed by our ears; this is why we use a logarithmic scale to better match our perception. In quiet rooms, sound intensity is low (around \( 10^{-10} \) W/m\( ^2 \)), while busy street corners can reach higher intensities such as \( 10^{-5} \) W/m\( ^2 \). The threshold of pain for our ears is about 1 W/m\( ^2 \), and it is fundamental to be aware of these levels for protecting our hearing health.

Properties of Logarithms

The properties of logarithms are mathematical rules we use to manipulate log expressions to make calculations possible or simpler. Some of these key properties come into play when solving sound intensity problems. One such property is the quotient rule \( \log_{b} \frac{a}{c} = \log_{b} a - \log_{b} c \), which helps simplify log expressions with division involved. Another property is the power rule that states \( \log_b(a^r) = r\log_b(a) \), and this is particularly useful when the argument of the log is an exponent.

Using these properties allows us to solve for decibel levels in the given exercise by transforming the logarithm expressions into simpler terms, providing a clear pathway from the intensity of sound to its perceived loudness in decibels.