Problem 35 - Precalculus Mathematics for Calculus

Question

The loudness \(L\) of a sound (measured in decibels, \(\mathrm{dB}\) ) is inversely proportional to the square of the distance \(d\) from the source of the sound. A person who is \(10 \mathrm{ft}\) from a lawn mower experiences a sound level of \(70 \mathrm{dB} .\) How loud is the lawn mower when the person is \(100 \mathrm{ft}\) away?

Step-by-Step Solution

Answer

The lawn mower is 0.7 dB at 100 ft away.

Step 1: Understand the Relationship

First, understand that the loudness \( L \) is inversely proportional to the square of the distance \( d \). This can be expressed as \( L = \frac{k}{d^2} \), where \( k \) is a constant of proportionality.

Step 2: Determine the Constant of Proportionality

Using the given condition, when \( d = 10 \) ft, \( L = 70 \) dB. Substitute these values into the formula to find \( k \): \[ 70 = \frac{k}{10^2} = \frac{k}{100} \]Multiply both sides by 100 to find \( k \): \[ k = 7000 \]

Step 3: Apply the Formula for a Different Distance

Now use the formula \( L = \frac{k}{d^2} \) with the new distance \( d = 100 \) ft to find the loudness \( L \): \[ L = \frac{7000}{100^2} = \frac{7000}{10000} \]Simplifying this gives: \[ L = 0.7 \]

Step 4: Conclusion

At 100 ft from the lawn mower, the loudness is \( 0.7 \) dB. This is due to the dramatic decrease resulting from inversely proportional square distance.

Key Concepts

Sound Intensity

Sound intensity refers to the power carried by sound waves per unit area in a direction perpendicular to that area. In simple terms, it's about how much sound energy is passing through a certain space at a given time. The intensity of sound helps us gauge how loud it actually is, and is measured in watts per square meter (W/m²).
Understanding sound intensity is important because it affects how we perceive loudness. As sound waves travel farther from the source, the energy spreads out over a larger area, making the sound seem quieter. This happens because energy gets dispersed, reducing the intensity and, as a result, the loudness of the sound.

• Sound intensity is measured in watts per square meter.
• It helps us understand how much energy a sound wave carries.
• Sound intensity decreases as distance from the source increases.

Essentially, the further you are from the source of the sound, the quieter it will seem because the energy is spread over a larger area.

Decibels

Decibels (dB) are the units used to measure the intensity or loudness of sound. It’s a logarithmic unit, meaning that a small change in decibels represents a large change in intensity. The decibel scale is used because the human ear perceives sound intensity changes logarithmically, allowing it to handle the vast range of sound levels experienced in the natural world.
For example, an increase of 10 dB represents a tenfold increase in intensity, meaning that 70 dB is ten times more intense than 60 dB. This scale helps us understand and compare different sounds easily, even if they span a wide range of intensities.

• Decibels are a logarithmic measure of sound intensity.
• A 10 dB increase represents a tenfold increase in sound intensity.
• They allow us to compare a wide range of sound levels easily.

When talking about sound and its distance from the source, decibels help calculate and understand how changes in distance affect perceived loudness.

Distance and Sound

The relationship between distance and sound is a fundamental concept in acoustics. As sound moves away from its source, its intensity decreases - this is often modeled using the concept of inverse proportionality. Specifically, the intensity is inversely proportional to the square of the distance from the sound source.
Imagine the sound waves spreading out in all directions from the source. With increasing distance, the same amount of sound energy has to cover more area, reducing its intensity at any one point. This principle explains why sounds grow quieter the further away you get.
The mathematical expression of this relationship is represented by the formula: \( L = \frac{k}{d^2} \), where:\( L \) is the loudness in decibels,
\( d \) is the distance from the sound source, and
\( k \) is a constant of proportionality.

• Sound intensity decreases as distance from the source increases.
• This relationship is modeled by inverse proportionality, \( L = \frac{k}{d^2} \).
• The principle explains why sounds appear quieter as you move further away.

Understanding this concept is crucial for predicting how loud or quiet a sound will be at different distances.