Problem 32 - Contemporary Precalculus

Question

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Ticking watch (intensity is 100 times \(i_{0}\) ).

Step-by-Step Solution

Answer

Answer: The loudness of the ticking watch is 20 decibels.

Step 1: Understand the given information

The energy intensity of the ticking watch is 100 times \(i_{0}\). We are given the function \(L(i)=10 \cdot \log \left(i / i_{0}\right)\), where L(i) is measured in decibels.

Step 2: Substitute the intensity and find the decibel measure

Since the intensity of the ticking watch is 100 times \(i_{0}\), we can substitute it into the equation as follows: \(L(i) = 10 \cdot \log \left((100 i_{0}) / i_{0}\right)\)

Step 3: Simplify and solve for L(i)

Simplify the equation by dividing the terms inside the logarithm. \(L(i) = 10 \cdot \log (100)\) Now, take the logarithm (the base of this logarithm is 10) of 100: \(\log (100) = 2.\) Thus, we get: \(L(i) = 10 \cdot 2\) \(L(i) = 20\) decibels The decibel measure of the sound produced by the ticking watch is 20 decibels.

Key Concepts

Sound Intensity

Sound intensity refers to the power carried by sound waves per unit area in a direction perpendicular to that area. The intensity of sound is a crucial aspect that helps in determining how loud we perceive a sound to be. The human ear can detect a very wide range of sound intensities, from the faint ticking of a watch to the loud roar of a jet engine.

- **Key Factors**: Sound intensity is influenced by the source of the sound and the medium through which it travels. - **Measurement**: It's measured in watts per square meter (W/m²).
Human hearing is particularly sensitive, and the ear can detect sounds with intensities as low as 0.000000000001 W/m². This minimum intensity that the human ear can detect is represented as \(i_0\). The intensity of sounds can vary greatly, so a convenient way of expressing these variances is by using the decibel scale.

Decibel Scale

The decibel scale is a logarithmic unit that is used to measure sound intensity levels. It allows us to describe a vast range of intensities in a more manageable format. This is because the human ear perceives sound intensity logarithmically rather than linearly.

- **Logarithmic Nature**: The decibel scale uses the logarithm to compare the intensity of a given sound to the reference intensity \(i_0\). This means when you hear a sound twice as intense, it's not perceived as twice as loud. Instead, it requires a much more significant change in intensity for it to sound "twice as loud."- **Calculation**: The formula provided, \(L(i)=10 \cdot \log \left(i / i_{0}\right)\), makes use of base 10 logarithms to convert intensity ratios into decibels.
For instance, the ticking of a watch at 100 times the reference intensity \(i_0\) results in \(L(i) = 10 \cdot \log(100) = 20\) decibels. This simple calculation demonstrates how logarithms make it easier to express varying sound intensities.

Logarithms in Real Life

Logarithms are not just mathematical abstract concepts but have real-world applications that make complex calculations more manageable. They are used in many areas such as sound measurement, as seen with the decibel scale, but also appear extensively in other domains.

- **Earthquake Intensity**: Logarithms are used in the Richter scale to measure the magnitude of earthquakes, where each whole number increase on the magnitude scale represents a tenfold increase in measured amplitude. - **pH Levels**: In chemistry, the pH scale is logarithmic, used to specify the acidity or basicity of an aqueous solution. - **Finance**: In finance, logarithms help in calculating compound interest and in several models assessing growth rates.
These examples show how logarithms simplify complex relationships, making them indispensable in analyzing and interpreting scientific data. By understanding logarithmic functions like those used with decibels, students can appreciate the broad applicability of these mathematical principles.