Problem 86 - College Algebra

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. Use the formula to solve Exercises. 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 the result obtained after solving the equation in step 3.

Step 1: Understand the Question

We are required to calculate the loudness of a sound given the intensity. We are provided with a formula to use for this calculation.

Step 2: Use the Formula

Substitute the given intensity \(I = 3.2 \times 10^{-6}\) into the formula \(D=10 \log \left(10^{12} I\right)\).

Step 3: Calculate

After substitution, the equation becomes: \(D=10 \log \left(10^{12} \times 3.2 \times 10^{-6}\right)\). Solving this equation gives the decibel level \(D\).

Key Concepts

Sound Intensity

Sound intensity refers to the amount of energy that sound waves carry per unit area in a direction perpendicular to that area.
It is measured in watts per square meter (W/m²). The higher the intensity, the louder the sound appears. When you think about sound intensity, imagine standing near a speaker at a concert.
The closer you are, the more energy, or "intensity," you experience. Sound intensity can range from very soft, like the rustle of leaves, to extremely loud, like a jet engine.
For our example, a normal conversation has an intensity of \(3.2 \times 10^{-6}\) W/m².

Logarithms in Algebra

Logarithms are a fundamental concept in algebra, often used to simplify complex multiplicative processes. A logarithm answers the question: "To what exponent must we raise a base number to obtain another number?"
Typically, base 10 logarithms are used in everyday contexts, especially concerning sound. When working with logarithms in sound equations, like in the decibel formula, they help us easily manage huge ranges of sound intensities on a reasonable scale.
The conversion of multiplication into addition via logarithms simplifies calculations. This is seen in the formula: \(\log(10^{12} \times I)\). By breaking it down:

• \(10^{12}\) represents a constant amplification factor, considering the threshold of human hearing.
• \(I\) is the specific sound intensity being measured.

Decibel Level Formula

The decibel scale is a logarithmic way to express the intensity of sounds compared to a reference level.
The formula used for computing decibel level from sound intensity is:\[D = 10 \log(10^{12} I)\]Here's how it works step-by-step:

• Substitute the sound intensity into the formula.
  For our example, \(I = 3.2 \times 10^{-6}\).
• Calculate the inside of the logarithm: \(10^{12} \times 3.2 \times 10^{-6}\).
• Find the logarithm of the result, transforming it to a manageable number.
• Multiply by 10 to get the decibel level \(D\).

Understanding this process allows us to translate physical sound phenomena into meaningful numbers that describe how loud or intense the sound actually is.
This method is crucial for various applications, from everyday noise assessments to professional audio engineering.