How does Brandon’s sound intensity level of 10^-10 compare to …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as L = 10 log StartFraction I Over I 0 EndFraction, where I 0 = 10 Superscript negative 12 and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is 10-10. Ahmad, Brandon’s neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is 10-4. How does Brandon’s sound intensity level compare to Ahmad’s mower?

Short Answer

Brandon’s sound level is significantly lower than Ahmad’s mower, with Brandon’s sound intensity at 10^-10 watts/m¬¨‚⧠compared to the mower’s 10^-4 watts/m¬¨‚â§. Therefore, Ahmad’s mower is 60 decibels louder than the sound level Brandon can hear, indicating a substantial difference in sound intensity.

Step-by-Step Solution

Brandon’s sound level comparison with Ahmad’s mower can be summarized in three clear steps.

Step 1: Understanding Sound Intensity and Decibels

Sound intensity is typically measured in terms of watts per square meter, and the relationship between sound intensity (I) and sound level in decibels (Db) is defined by the formula L = 10 log (I/I0). Here, I0 is the reference sound intensity, which equals 10^-12 watts/m², representing the threshold of hearing for the average human ear. Understanding this foundation is critical for comparing different sound intensities.

Step 2: Measuring Brandon and Ahmad’s Sound Levels

Brandon’s hearing capability is defined at a sound intensity of 10^-10 watts/m¬¨‚â§, while Ahmed’s mower generates a much higher intensity of 10^-4 watts/m¬¨‚â§. To compare these levels accurately, we calculate the ratio of intensities, which shows that Ahmad’s mower is 10^6 (one million times) more intense than the sound Brandon can perceive. This significant difference indicates that the mower operates at a much higher sound intensity level.

Step 3: Calculating the Decibel Difference

Using the logarithmic nature of the decibel scale, we establish that every increase of tenfold in sound intensity results in a 10 decibel increase in loudness. Given that Ahmad’s mower is 10^6 times more intense than Brandon’s sound level, we can apply this relationship:

  • The increase in decibels is calculated as follows: 10 log(10^6) = 60 decibels.
  • This means Ahmad’s mower is 60 decibels louder than the sound level Brandon can hear.

Related Concepts

Sound Intensity

The power per unit area carried by a sound wave, typically measured in watts per square meter

Decibels

The logarithmic unit used to measure the intensity of a sound, where each increase of 10 db represents a tenfold increase in sound intensity

Threshold Of Hearing

A reference sound intensity level of 10^-12 watts/m², which represents the quietest sound that the average human ear can perceive.

Table Of Contents
  1. The loudness, L, measured in decibels (Db), of a sound intensity, I, measured in watts per square meter, is defined as L = 10 log StartFraction I Over I 0 EndFraction, where I 0 = 10 Superscript negative 12 and is the least intense sound a human ear can hear. Brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. The sound intensity level that Brandon can hear is 10-10. Ahmad, Brandon’s neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is 10-4. How does Brandon’s sound intensity level compare to Ahmad’s mower?

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