Problem 118 - College Algebra Essentials

Question

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where \(I\) is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a ruptured eardrum. (Any exposure to sounds of I3 0 decibels or higher puts a person at immediate risk for hearing damage.) What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Step-by-Step Solution

Answer

The decibel level of a normal conversation is 60.

Step 1: Identify Given and Required Variables

We are given the intensity \(I = 3.2 \times 10^{-6}\) watt per meter^2. We are required to find the decibel level \(D.\)

Step 2: Apply Formula

Now we apply the given formula \(D = 10\log(10^{12} I)\) by substituting the value of \(I\) into it.

Step 3: Calculate Decibel Level

This gives us \(D = 10\log(10^{12} \times 3.2 \times 10^{-6})\). Simplifying further we get \(D = 10\log(3.2 \times 10^{6})\). Evaluating the logarithm in this formula gives us the output \(D = 60\) decibels.

Key Concepts

Logarithmic Function

The logarithmic function is a mathematical tool that is essential in many fields, including sound measurement. In simple words, a logarithm is the opposite of exponentiation. When you have a log function, it means you're trying to find out how many times you multiply a certain number (called the base) to get another number. For instance, in the equation \( D = 10 \log(10^{12} I) \), the base of the log is 10. This means we're dealing with a logarithm base 10, which is quite common. This particular function takes a product (in this case, sound intensity times a large constant) and turns it into a simpler measure.

• Logarithms help simplify large ranges into manageable numbers.
• This also explains why logs are used in the decibel calculation.

When dealing with sound, which can range from almost zero to very powerful, logs help convert that large range into a decibel scale that's easier to work with.

Sound Intensity

Sound intensity is a measure of how much energy a sound wave carries per unit area. It's quantified in terms of watts per meter squared (\( \text{W/m}^2 \)). In simpler terms, sound intensity indicates how "strong" or "powerful" a sound is.For a clearer understanding, think of sound intensity as the "volume knob," if you will, that tells how loud a sound is based on the energy it transmits. When you are measuring the sound intensity, you are essentially measuring the energy exposure to the ear.

• Sound intensity is often used to calculate decibel levels, which are then used to express loudness.
• This intensity has a huge range from whispering (very low) to a jet engine (very high).

Loudness Level

The loudness level of a sound is a subjective measure perceived by human ears. While it relates to sound intensity, it isn't the same because human perception of loudness is not linear with intensity. Measurements of loudness level rely on the decibel scale, which is logarithmic in nature. The logarithmic relationship means that each increase of 10 decibels represents a tenfold increase in intensity, but only a doubling of perceived loudness.

• This explains why a normal conversation, calculated to be at 60 decibels, feels quite audible but not overwhelmingly loud.
• Loudness is an essential consideration in settings where noise regulation is necessary.

Understanding loudness level helps in designing better auditory environments and protecting hearing health.

Decibel Scale

The decibel (dB) scale is a logarithmic scale used to measure sound levels. This scale makes it easier to express the incredibly vast range of sounds that the human ear can detect, from the faintest whisper to the roar of a rocket. The scale is logarithmic because the ear perceives sound intensity in a nonlinear fashion. A logarithmic scale reflects the way humans experience changes in sound intensity. This is why the decibel scale is a critical concept in fields like acoustics.

• Each 10 dB increase represents a doubling of loudness.
• Acoustic events like normal conversation are at 60 dB, while sounds above 130 dB risk hearing damage.

Therefore, using the decibel scale helps in quantifying sound in a way that aligns with human auditory perception, enabling effective communication of sound levels.