Problem 535 - 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 large orchestra with a sound intensity of 6.3\(\cdot 10^{-3}\) watts per square meter?

Step-by-Step Solution

Answer

The large orchestra emits approximately 98 decibels.

Step 1: Identify the given values

We are given the sound intensity \( I \) of a large orchestra as \( I = 6.3 \cdot 10^{-3} \) watts per square meter. Also, \( I_{0} \) is given as \( I_{0} = 10^{-12} \) watts per square meter, which is the reference intensity.

Step 2: Substitute values into the formula

Substitute \( I = 6.3 \cdot 10^{-3} \) and \( I_{0} = 10^{-12} \) into the decibel equation:\[D = 10 \log\left(\frac{I}{I_{0}}\right) = 10 \log\left(\frac{6.3 \cdot 10^{-3}}{10^{-12}}\right)\]

Step 3: Simplify the fraction inside the logarithm

Calculate the fraction \( \frac{6.3 \cdot 10^{-3}}{10^{-12}} \):\[\frac{6.3 \cdot 10^{-3}}{10^{-12}} = 6.3 \cdot 10^{9}\]This simplification is achieved by dividing the powers of ten.

Step 4: Calculate the logarithm

Now calculate the logarithm:\[\log(6.3 \cdot 10^{9}) = \log(6.3) + \log(10^{9})\]

Step 5: Use the properties of logarithms

Calculate each part separately:\[ \log(10^{9}) = 9\]Use a calculator or logarithm table for \( \log(6.3) \), approximately \( \log(6.3) \approx 0.799 \), so:\[\log(6.3 \cdot 10^{9}) = 0.799 + 9 = 9.799\]

Step 6: Calculate the decibels

Now substitute back into the formula:\[D = 10 \cdot 9.799 = 97.99 \]This means the sound intensity is approximately 98 decibels.

Key Concepts

Decibels

Decibels, often abbreviated as dB, are the units used to measure sound intensity levels. A decibel quantifies sound intensity based on a logarithmic scale, which means it compares not absolute values but ratios. Decibels help us understand sound levels because human hearing perceives sound increases in a logarithmic manner.

In simple terms, every 10-decibel increase means the sound is perceived as about twice as loud. This scale enables us to handle a wide range of sound intensities—from the softest whisper to the loudest explosion—in a compact form. The decibel scale is fundamental in a wide range of applications, from audio technology to environmental noise measurement.

Logarithmic Scale

A logarithmic scale is a type of scale used for a large range of quantities. In contrast to a linear scale, where values increase at constant intervals, a logarithmic scale increases in powers of ten. This is the reason why the decibel scale is logical—not only for articulating sound intensity but also for expressing very large or small numbers efficiently.

In the context of sound, our hearing works on this logarithmic principle. Small increases in decibels represent much larger increases in actual sound pressure levels. This type of scaling makes it easier to manage and visualize data that involve multiplicative relationships.

Sound Intensity Formula

The sound intensity formula relates intensity and decibels and provides a way to convert between linear intensity units (watts per square meter) and decibels, a logarithmic unit. The formula is given by: \[ D = 10 \log \left( \frac{I}{I_0} \right) \] where \( I \) is the intensity of the sound, and \( I_0 \) is the reference intensity, often the threshold of human hearing, \( 10^{-12} \) watts per square meter.

This formula follows from the idea that the power of sound increases exponentially while the human ear perceives it linearly. Thus, converting an intensity to decibels using this formula allows us to express how humans actually experience sound.

Properties of Logarithms

Understanding the properties of logarithms aids in simplifying calculations involving sound intensity. Some key properties include:

• \( \log(ab) = \log(a) + \log(b) \)
• \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \)
• \( \log(a^b) = b \log(a) \)

In the problem of finding sound intensity in decibels, these properties allow us to break down complex multiplication or division inside a logarithm into simpler steps. For instance, converting \[ \log(6.3 \cdot 10^{9}) \] into \[ \log(6.3) + 9 \] uses the property \( \log(ab) = \log(a) + \log(b) \). These properties make logarithmic expressions more manageable and help in understanding how changes in intensity relate to changes in decibels.