Problem 581 - Precalculus

Question

The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10 \log \left(\frac{1}{I_{0}}\right),\) where \(I\) is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of 4.7\(\cdot 10^{-1}\) watts per square meter?

Step-by-Step Solution

Answer

The sound intensity from the rock concert is 116.7 decibels.

Step 1: Identify the given values

We're given the sound intensity \( I \) at a rock concert as \( 4.7 \times 10^{-1} \) watts per square meter and \( I_0 = 10^{-12} \) watts per square meter, the threshold of human hearing.

Step 2: Substitute into the formula

Substitute \( I = 4.7 \times 10^{-1} \) and \( I_0 = 10^{-12} \) into the formula \( D = 10 \log \left(\frac{I}{I_0}\right) \). The formula becomes \( D = 10 \log \left(\frac{4.7 \times 10^{-1}}{10^{-12}}\right) \).

Step 3: Simplify the fraction inside the logarithm

Simplify \( \frac{4.7 \times 10^{-1}}{10^{-12}} \). This is equivalent to \( 4.7 \times 10^{-1 + 12} = 4.7 \times 10^{11} \).

Step 4: Calculate the logarithm

Calculate \( \log (4.7 \times 10^{11}) \) using the property \( \log(ab) = \log(a) + \log(b) \). So, it becomes \( \log(4.7) + \log(10^{11}) \).

Step 5: Use logarithmic properties

Since \( \log(10^{11}) = 11 \), the expression simplifies to \( \log (4.7) + 11 \).

Step 6: Find the numerical value of the logarithm

Using a calculator, find \( \log(4.7) \approx 0.6721 \). So, the expression becomes \( 0.6721 + 11 = 11.6721 \).

Step 7: Final calculation of decibels

Multiply the result by 10, as in the original formula: \( D = 10 \times 11.6721 = 116.721 \).

Step 8: Round the result

Round 116.721 to one decimal place or as required by the context: 116.7 decibels.

Key Concepts

Logarithmic Functions

Logarithmic functions play a crucial role in various scientific and mathematical calculations, especially when dealing with exponential phenomena like sound intensity. The logarithm in base 10, commonly denoted as \( \log \), represents the power to which the base 10 must be raised to obtain a particular number. For example, \( \log(100) = 2 \) because \( 10^2 = 100 \). In the context of sound intensity, the logarithmic function helps express a wide range of sound intensities in a more manageable scale, the decibel scale, which uses logarithms to equate differences in levels of power or intensity.

The decibel scale compresses a vast array of sound intensities into a more easily understandable format, allowing us to comprehend how loud something is in a real-world context. This is useful because the human ear perceives sound intensity non-linearly, which aligns with the logarithmic nature of the decibel scale. This function allows for simplifying and working with the properties of sound in ways that are more intuitive for human perception.

Intensity of Sound

The intensity of sound is a measure of the power conveyed by a sound wave per unit area, and it's usually expressed in watts per square meter. In our exercise, we use a given sound intensity value, \( I \), which is indicative of how much energy the sound wave carries. This intensity is crucial because it directly correlates with how loud the sound will be perceived to be.

Sound intensity is often measured with respect to the minimum threshold of hearing, represented by \( I_0 = 10^{-12} \) watts per square meter. This threshold is a reference point, meaning if a sound's intensity is \( 10^{-12} \), it is at the lowest limit of audibility for the average human ear.
The transformation of this physiological parameter into the more intuitive decibel units using the logarithmic equation \( D = 10 \log \left( \frac{I}{I_0} \right) \) makes it feasible to convey real-world sound experiences quantitatively. It also makes calculations and communication about sound intensity more accessible to non-specialists.

Properties of Logarithms

Understanding the properties of logarithms is key to simplifying and solving equations involving sound intensity measured in decibels. Some pivotal properties include:

• **Product Property**: \( \log(ab) = \log(a) + \log(b) \)
• **Quotient Property**: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)
• **Power Property**: \( \log(a^b) = b \cdot \log(a) \)

In our exercise, we utilized these properties, especially when simplifying the logarithmic calculation \( \log(4.7 \times 10^{11}) \). By applying the product property, we split the expression into two components: \( \log(4.7) + \log(10^{11}) \).

Furthermore, recognizing that \( \log(10^{11}) = 11 \) simplifies the process. The ability to quickly transition from complex expressions to simpler arithmetic is what makes logarithms so useful in converting between linear and exponential scales. It's these properties that help decode the exponential curve of sound intensity into the linearly comprehensible decibel format.