Problem 356 - Precalculus

Question

Solve for the indicated value, and graph the situation showing the solution point. The formula for measuring sound intensity in decibels \(D\) is decined 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 plane emits approximately 149.19 decibels.

Step 1: Identify the Given Values

We are given that the sound intensity \(I\) is 8.3 \( \times 10^2 \) watts per square meter, and \(I_{0}\) is \(10^{-12}\) watts per square meter. Our task is to find the decibel level \(D\) using the formula \(D=10 \log \left(\frac{I}{I_{0}}\right)\).

Step 2: Substitute the Values into the Formula

Insert the values of \(I = 8.3 \times 10^2\) and \(I_0 = 10^{-12}\) into the equation. This gives us:\[D = 10 \log \left(\frac{8.3 \times 10^2}{10^{-12}}\right)\]

Step 3: Simplify the Expression Inside the Logarithm

Calculate the ratio \(\frac{8.3 \times 10^2}{10^{-12}}\) by using the property of exponents \(a^m / a^n = a^{m-n}\): \[\frac{8.3 \times 10^2}{10^{-12}} = 8.3 \times 10^{2 + 12} = 8.3 \times 10^{14}\]

Step 4: Calculate the Logarithm

Compute \(\log (8.3 \times 10^{14})\) using logarithmic properties.\[\log (8.3 \times 10^{14}) = \log 8.3 + \log 10^{14} = \log 8.3 + 14\]Using a calculator, \(\log 8.3 \approx 0.919\), so:\[\log (8.3 \times 10^{14}) \approx 0.919 + 14 = 14.919\]

Step 5: Calculate the Decibels

Multiply the result from Step 4 by 10 to solve for \(D\):\[D = 10 \times 14.919 = 149.19\]So, the jet plane emits approximately 149.19 decibels.

Step 6: Graph the Situation

To graph the situation, use a graph with the x-axis representing sound intensity in watts per square meter and the y-axis representing decibels. Plot the point (8.3 \( \times 10^2 \), 149.19) to show the sound intensity and corresponding decibel level of the jet plane.

Key Concepts

Understanding Decibels

Decibels (dB) are a unit for measuring sound intensity. They provide a way to express the power or intensity of sound in a manner that is easier to understand and compare. Sound intensity is expressed in watts per square meter, but using decibels helps to accommodate the vast range of sounds the human ear can perceive.

• Decibels are calculated using a logarithmic scale, which means they represent powers of 10. This makes it simpler to compare large ranges of sound intensities.
• The formula for calculating decibels is: \(D = 10 \log \left(\frac{I}{I_0}\right)\), where \(I_0 = 10^{-12}\) is the threshold of human hearing, the quietest sound most humans can hear.
• Decibels provide a relative measure of power level, allowing you to understand how much more intense one sound is compared to another.

When you think about decibels, remember that each 10 dB increase represents a tenfold increase in intensity. So, a sound at 40 dB is 10 times more intense than at 30 dB, and 100 times more intense than at 20 dB.

Logarithmic Equations in Sound

Logarithmic equations are central to understanding sound intensity and decibel calculations. Logarithms help translate complex multiplicative processes into more manageable additive ones.

• In the equation \(D = 10 \log \left(\frac{I}{I_0}\right)\), logarithms allow us to manage large variations in sound intensities efficiently.
• By expressing \(I/I_0\) in terms of a logarithm, we simplify the process of scaling large numbers down to a smaller, more interpretable range.
• Logarithmic equations, like this one, enable us to quantify and compare sound levels in a more intuitive way, since our hearing perceives changes in intensity logarithmically.

Understanding how to work with logarithms is crucial because it allows you to rearrange the formula and solve for unknowns, whether they are inside the logarithm or multiplied with it. This makes logarithmic equations extremely versatile.

Graphing Sound Intensity Functions

Graphing functions that depict sound intensity help us visualize relationships between sound intensity and decibels.

• The graph uses the x-axis to represent sound intensity (in watts per square meter) and the y-axis for decibel levels.
• Plotting points like (8.3 x 10^2, 149.19) demonstrates how intense sounds, like a jet plane, relate to decibel measurements.
• A curve on this graph illustrates how increases in intensity impact the decibel level, showcasing the logarithmic nature of the scale.

By graphing these functions, you can visually interpret the exponential growth in sound intensity relative to the decibel level, emphasizing how a small change in input intensity can result in significant changes in perceived loudness. Graphing helps not just with analysis but also with predicting and experimenting with sound intensities in a real-world context.