Problem 91 - Introductory and Intermediate Algebra for College Students 4th

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 nuptured eardrum. Use the formula to solve Exercises. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?

Step-by-Step Solution

Answer

The decibel level of the sound of a blue whale is approximately 180 decibels, which exceeds the 160-decibel maximum level the human eardrum can sustain without being ruptured. Therefore, at close range, the sound of a blue whale can rupture the human eardrum.

Step 1: Identify the given intensity

The intensity of the sound of a blue whale is given as \(6.3 \times 10^{6}\) watts per meter squared.

Step 2: Substitute the intensity into the formula

Substitute the given sound intensity \(I\) into the formula for loudness \(D\). This gives us \(D=10 \log(10^{12} \times (6.3 \times 10^{6}))\).

Step 3: Evaluate the expression

By calculating this expression, the loudness level \(D\) will be obtained in decibels. Remember that when multiplying numbers with the same base, the exponents are added. Thus, the formula can be simplified to \(D=10 \log(10^{18})\), which is approximated to 180 decibels.

Step 4: Compare the result with the maximum non-rupture decibel level

A sound that reaches a loudness level of 160 decibels can rupture a human eardrum. Therefore, since 180 decibels (the loudness level of the blue whale) is higher than 160 decibels, this indicates that at close range, the sound of a blue whale can rupture the human eardrum.

Key Concepts

Sound Intensity

Sound intensity is a measure of how much power a sound carries over an area. It's usually measured in watts per meter squared. This value represents how "strong" or "loud" a sound feels.
A sound, like the one from a blue whale, can have an amazingly high intensity, making it travel vast distances. For instance, the given sound intensity of a blue whale is 6.3 million watts per meter squared or \(6.3 \times 10^{6}\) watts.

Such high-intensity values are often experienced in nature. Understanding sound intensity is crucial, especially since it allows us to comprehend how sound propagates in various environments, and its potential impact on human health.

• Higher sound intensity means more powerful sound energy per unit area.
• It's important for determining how far a sound can travel and how it affects listeners.

Grasping this concept informs how different sound levels can affect our health and environment.

Decibels Calculation

Decibels are the units used to measure the intensity of a sound. They provide a way to express large variations in sound intensity in a manageable format. The formula for calculating decibels involves logarithmic functions, reflecting the exponential nature of sound intensity perception.
To calculate the decibel level, the formula used is:
\[D=10 \log \left(10^{12} \times I\right)\]Here's how decibels are calculated:

• The intensity \(I\) is first multiplied by \(10^{12}\). This scales the intensity to a comparable range of human hearing sensitivity.
• Then, take the logarithm (base 10). Logarithms simplify large numbers, converting multiplicative factors into additive ones.
• Finally, multiply by 10 to adjust the scale.

In our case, substituting the blue whale's intensity gives:\(D=10 \log(10^{12} \times 6.3 \times 10^{6}) = 180\) decibels. This result demonstrates that the sound is extremely loud and can indeed cause damage at close range.

Exponents in Logarithms

Exponents and logarithms are mathematical tools that transform complex multiplicative relationships into simpler additive ones. They are essential in understanding calculations involving large values, such as sound intensities.
Logarithms work as the inverse of exponential functions, meaning they help find the exponent that a base number needs to be raised to in order to produce another number. For example, in the context of sound intensity:
The use of \(10^{12}\) in decibel calculations relates to converting sound intensity into an easily comparable scale.

• Assume an intensity \(I\), such as \(6.3 \times 10^{6}\), and multiply by base \(10^{12}\).
• The formula simplifies to \(D=10 \log(10^{12} \times (6.3 \times 10^{6}))\), making it easier to calculate using basic logarithm properties.
• The property of exponents and logarithms means that \(D=10 \log(10^{18})\) can be directly calculated as \(180\) by adding the exponents.

Understanding these properties is key to solving problems involving powers and logarithms, especially in fields like acoustics and electronics.