91Ó°ÊÓ

Decibels: Sound exerts a pressure \(P\) on the human ear. This pressure increases as the loudness of the sound increases. It is convenient to measure the loudness \(D\) in decibels and the pressure \(P\) in dynes per square centimeter. It has been found that each increase of 1 decibel in loudness causes a \(12.2 \%\) increase in pressure. Furthermore, a sound of loudness 97 decibels produces a pressure of 15 dynes per square centimeter. a. Explain why \(P\) is an exponential function of \(D\) and find the growth factor. b. Find \(P(0)\) and explain in practical terms what your answer means. c. Find an exponential model for \(P\) as a function of \(D\). d. When pressure on the ear reaches a level of about 200 dynes per square centimeter, physical damage can occur. What decibel level should be considered dangerous?

Step by step solution

01

Determining the Exponential Relationship

The given problem states that a 1 decibel increase causes a 12.2% increase in pressure. An exponential function is characterized by consistent percentage growth per unit increase in the independent variable. Hence, the pressure \(P\) as a function of loudness \(D\) is exponential.

02

Calculating the Growth Factor

For an increase of 12.2% per decibel, the growth factor per decibel is \(1 + 0.122 = 1.122\). Thus, the pressure increases by a factor of 1.122 for each decibel increase.

03

Finding Initial Pressure P(0)

Given that a loudness of 97 decibels corresponds to 15 dynes/cm², we use the decay factor\( \left( \frac{1}{1.122}\right) \) to calculate \(P(0)\):\[ P(D) = 15 \, \text{when} \, D = 97 \]\[ P(0) = 15 \times \frac{1}{1.122^{97}} \]Calculating gives \( P(0) \approx 0.01807 \ \, \text{dynes/cm}^2 \). This implies the pressure corresponding to 0 decibels is extremely low.

04

Constructing the Exponential Model

Using the formula \(P(D) = P_0 \times r^{D}\), where \(r = 1.122\) and \(P_0 = 0.01807\), the model is:\[ P(D) = 0.01807 \times 1.122^D \]

05

Calculating Dangerous Decibel Level

To find the dangerous decibel level \(D\) when \(P = 200\):\[ 200 = 0.01807 \times 1.122^D \]First, resolve \(1.122^D = \frac{200}{0.01807}\), then solve for \(D\) by taking the logarithm:\[ D = \frac{\log(11069.45)}{\log(1.122)} \]Calculating gives \(D \approx 121\). Thus, 121 decibels should be considered dangerous.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decibels

When we hear sound, we're actually experiencing pressure waves traveling through the air, and our ears detect them. The term "decibels" pertains to how we measure the **loudness** of a sound. Instead of gauging sound in mere 'units,' which might be cumbersome due to its wide-ranging nature, decibels offer us a simpler scale. The decibel scale is logarithmic, which means each increase of one decibel equates to a noticeable increase in the sound pressure. In technical terms, the scale helps us express large numbers in a more manageable way, using powers of ten. If you see a 12.2% increase in pressure with each additional decibel, it implies that this unit consistently reflects changes in loudness through exponential growth. By understanding this, you grasp why decibels are paramount for illustrating sound loudness.

Sound Pressure Level

Sound exerts pressure on our ears. **Sound pressure level** is a way to quantify this pressure. It's measured in dynes per square centimeter. Consider sound pressure level like the force per unit area the sound exerts. For example, a normal conversation might produce a sound pressure level of around 60 decibels, while a jet engine might be near 140 decibels. The sound pressure level not only defines how loud something is but also how it impacts our hearing. A continuous or intense sound pressure beyond certain limits, such as 200 dynes per cm², can damage hearing structures, leading to a condition known as noise-induced hearing loss. This connection underscores why monitoring sound pressure levels is crucial.

Mathematical Modeling

**Mathematical modeling** involves creating equations or formulas to represent real-world phenomena. With sound, we model how the pressure changes with loudness using exponential functions. The key is to understand how a small increase on one scale results in a significant rise on another. By setting up the model, like for loudness and pressure, it becomes possible to forecast outcomes and predict future values. These models, based on exponential characteristics (like the 1.122 growth factor), serve as excellent tools for providing deeper insights. Realistic predictions can be made, such as the calculation that approximately 121 decibels is hazardous, showing the power of mathematical modeling.

Exponential Growth

**Exponential growth** describes a process by which quantities increase over time, where the rate of growth is proportional to the current value. In our exercise, sound pressure shows exponential growth with each decibel increase. This type of growth is common in fields like finance and population studies but is also essential in acoustics. In the context of sound, the pressure grows by each decibel's percentage increase, specifically 12.2%. To visualize, picture a snowball rolling down a hill, picking up more snow, representing how sound pressure increases rapidly as decibels add up. This is why exponential functions are instrumental; they convey how such rapid changes happen. By understanding exponential growth, you gain clarity on why even small increases in decibel level significantly impact our hearing.