Problem 34 - 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. Loud conversation (intensity is 4 million times \(i_{0}\) ).

Step-by-Step Solution

Answer

Answer: The decibel measure of the sound is approximately 66.02 dB.

Step 1: Write the given function and substitution

We have the function \(L(i)=10 \cdot \log \left(i / i_{0}\right)\). We are given that \(i = 4 \cdot 10^6 \cdot i_0\), which means \(\frac{i}{i_0} = 4 \cdot 10^6\).

Step 2: Substitute the value into the function

Now, we can substitute the value of \(\frac{i}{i_0}\) into the function: \( L(i) = 10 \cdot \log \left( 4 \cdot 10^6 \right) \).

Step 3: Simplify the logarithm

To simplify, we can use the logarithm properties, which state that \(\log(a \cdot b) = \log(a) + \log(b)\) and \(\log(a^k) = k \cdot \log(a)\). Therefore, \( L(i) = 10 \cdot \left(\log \left( 4 \right) + \log \left( 10^6 \right)\right) \).

Step 4: Calculate the decibel measure

We know that \(\log \left( 10^k \right) = k\). Therefore, \( L(i) = 10 \cdot \left(\log \left( 4 \right) + 6\right) \). Now, you can use a calculator or logarithm table to find the value of \(L(i)\): \( L(i) = 10 \cdot (\approx 0.60206 + 6) \)

Step 5: Final result

After calculating, we get: \( L(i) \approx 66.0206 \) So the decibel measure of the loud conversation is approximately 66.02 dB.

Key Concepts

Sound Intensity

Sound intensity is a concept that refers to the amount of energy that sound waves carry per unit area per unit time. It determines how loud a sound will be perceived. The more intense a sound, the louder it appears to human ears. Sound intensity is measured in watts per square meter (W/m²). The human ear can detect a wide range of intensities, from the faintest whisper to the loudest explosion.

In everyday life, typical sound sources include:

• A whisper (very low intensity)
• Normal conversation (moderate intensity)
• A concert or a jet engine (high intensity)

The minimum sound intensity that the average human ear can detect is known as the reference intensity, denoted by \( i_0 \). In terms of calculations, \( i_0 \) is often used as a baseline in logarithmic functions for sound measurement, allowing us to compare different sound intensities effectively.

Decibels

Decibels (dB) are a unit of measurement used to express sound intensity levels. The decibel scale is logarithmic, meaning each increase of 10 dB represents a tenfold increase in intensity. This makes it a very practical measurement for comparing large ranges of sounds.

A few key points about decibels:

• 0 dB is the threshold of hearing, the quietest sound that a human can typically hear.
• 10 dB represents sound that is ten times more intense than 0 dB.
• 30 dB might be equivalent to a quiet library.
• 60-70 dB is akin to normal conversation levels.
• 120 dB can feel like standing near a jet engine and may cause pain or damage to hearing.

In the context of the textbook problem, when a sound intensity is expressed as a multiple of \( i_0 \), the reference intensity, we can use the decibel formula \( L(i) = 10 \cdot \log(i / i_0) \) to find the loudness in decibels.

Logarithm Properties

Logarithm properties help simplify complex expressions and solve equations where exponential relationships exist. They are particularly useful in sound intensity calculations as the intensity levels are often expressed with large numbers due to their exponential nature.

Important logarithm properties include:

• The product rule: \( \log(a \cdot b) = \log(a) + \log(b) \)
• The power rule: \( \log(a^k) = k \cdot \log(a) \)
• The quotient rule: \( \log(a / b) = \log(a) - \log(b) \)
• The change of base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \( c \) is the new base.

By applying these rules, we can break down complex logarithmic expressions into simpler parts. In the given exercise, the logarithm properties were used to simplify the expression involving \( 4 \cdot 10^6 \) into a more manageable calculation: \( \log(4) + \log(10^6) \). This enabled a straightforward calculation of the decibel measurement.

Mathematical Modeling

Mathematical modeling is a method of using mathematical concepts and language to represent real-world phenomena. It involves constructing a mathematical framework that captures essential features of real situations and allows predictions and analysis.

In the context of sound and decibels, mathematical modeling helps us understand how sound intensity relates to perceived loudness. The formula \( L(i) = 10 \cdot \log(i / i_0) \) is a model that describes this relationship using logarithms with respect to a reference intensity. It allows scientists, engineers, and even in daily applications, people to predict how changing sound intensity influences perceived volume.

These models simplify complex natural processes to understandable formulas, aiding in fields like acoustics, audio engineering, and hearing sciences. Such modeling is crucial for designing everything from concert halls to high-quality headphones, ensuring sound is reproduced accurately and comfortably for human listeners.