Problem 66 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. The formula for measuring sound intensity in decibels \(D\) is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right),\) 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 jet plane with a sound intensity of \(8.3 \cdot 10^{2}\) watts per square meter?

Step-by-Step Solution

Answer

The jet emits 149.19 decibels.

Step 1: Identify the given values

We are given the sound intensity \( I = 8.3 \times 10^{2} \text{ watts/m}^2 \) and the reference intensity \( I_0 = 10^{-12} \text{ watts/m}^2 \). Our task is to find the decibel level \( D \) using the formula \( D = 10 \log \left(\frac{I}{I_0}\right) \).

Step 2: Setup the formula for D

Substitute the given values into the formula:\[D = 10 \log \left( \frac{8.3 \times 10^2}{10^{-12}} \right)\]

Step 3: Divide the intensities

Calculate the fraction inside the logarithm:\[\frac{8.3 \times 10^2}{10^{-12}} = 8.3 \times 10^{14}\]

Step 4: Calculate the logarithm

Find the logarithm of the result:\[\log(8.3 \times 10^{14}) = \log(8.3) + \log(10^{14})\]Using properties of logarithms:\[\log(10^{14}) = 14\]Calculate \(\log(8.3)\) using a calculator (approximately 0.919):\[\log(8.3 \times 10^{14}) \approx 0.919 + 14 = 14.919\]

Step 5: Calculate D

Now multiply the logarithm value by 10 to get \( D \):\[D = 10 \times 14.919 = 149.19\]

Step 6: Graph the solution

To represent the solution on a graph, plot sound intensity on the x-axis and decibel level \(D\) on the y-axis. Plot the point \((8.3 \times 10^2, 149.19)\) to illustrate the decibel level for the given intensity. This shows the relationship between sound intensity and decibel level.

Key Concepts

Sound Intensity

Sound intensity is the power per unit area carried by a sound wave and is an essential concept in understanding acoustics and how we interpret sound. It is measured in watts per square meter (W/m²). The reference intensity, denoted by \( I_0 \), is the baseline used in calculations for decibels, often set at \( 10^{-12} \text{ watts/m}^2 \). This value represents the quietest sound the average human ear can detect.
The intensity of sound determines its loudness, but our ears perceive changes in loudness logarithmically rather than linearly. This perception is why the decibel scale, a logarithmic scale, is used. Understanding sound intensity helps us navigate environments with various sounds, balancing between what we can tolerate and what might be harmful.

Logarithmic Functions

Logarithmic functions are an integral part of calculating decibels, as the intensity of sound is compared to a reference level using these functions. In this context, the logarithmic equation used is:

• \( D = 10 \log \left( \frac{I}{I_0} \right) \)

Here, \( \log \) denotes the common logarithm (base 10). Because human hearing varies exponentially with intensity, logarithms simplify the wide range of intensities into a more manageable scale.
This function translates large numerical ranges into smaller, simpler numbers while maintaining the proportional relationships between them. Logarithmic functions have unique properties, such as transforming multiplications into additions, useful when working with sound intensities and other exponential growths.

Graphing Solutions

Graphing solutions involves visually representing the relationship between sound intensity and decibels. A graph can be particularly helpful to understand how even small increases in intensity can lead to significant changes in decibels due to the logarithmic nature of the calculation.
In graphing the given exercise, the x-axis represents the sound intensity (\( 8.3 \times 10^{2} \text{ watts/m}^2 \)), and the y-axis represents the decibels calculated \( D = 149.19 \). The point \((8.3 \times 10^2, 149.19)\) is plotted to show the specific relationship between the provided sound intensity and its decibel level. Observing such graphs can enhance comprehension by providing visual insights into potentially abstract mathematical concepts.