Question

What is the decibel level of the sound of a monster truck with intensity 10−2 watts per square inch? Use a logarithmic model to solve.

Answer 1

Answer:

BdB = 68.10 decibels

Step-by-step explanation:

BdB = 10 log10 I / Io

where BdB is the sound intensity level in decibels, I is the sound intensity on the linear scale (W / m²) and I0 is the hearing threshold (10^-12 W / m²).

Converting Io de 10^-12 W/m² to W/in², tenemos

10^-12 W/m² = 1.5500031000062x10^-9 W/in²

So, applying the equation or formula,

BdB = 10 log10 (10 ^ -2 W/in² / 1.5500031000062x10^-9 W/in² )

BdB = 10 log10 (0.64516x10^6)

BdB = 10 log10 (6,451,600)

Being log10 (6,451,600) = 6.8096674332398761189214526331036, then

BdB = 10 x 6.8096674332398761189214526331036

BdB = 68.096674332398761189214526331036

BdB = 68.10 decibels

Answer 2

The decibel level of the sound of a monster truck is $\fbox{\begin\\\ \beta(db)=68.1\\\end{minispace}}$

Further explanation:

Decibel is defined as a unit which is used to measure the relative loudness or intensity of the sound.  

In the question it is given that the intensity of the sound of a monster truck is $10^{-2}$ watts per square inch.  

The formula which is used to calculate the decibel level of a sound is as follows:  

$\fbox{\begin\\\ \beta(db)=10\text{log}\left(\dfrac{I}{I_{0}}\right)\\\end{minispace}}$  

Here, $I$ represents the intensity of the sound of a monster truck and $I_{0}$ represents the threshold intensity of hearing at $1000\text{hz}$.  

The value of the threshold intensity of hearing at $1000\text{hz}$ is $10^{-12}$ watts per square meter.  

This implies that the valiue of $I_{0}$ is $10^{-12}$ watts per square meter.  

The intensity of the sound of a monster is given in unit of per square inch.  

The relation between per square inch and per square meter is as follows:    

$\fbox{\begin\\\ 1\text{square inch}=\dfrac{1}{1550}\text{square meter}\\\end{minispace}}$

So, the intensity of a monster truck is converted into per square meter as follows:  

$10^{-12}\text{watts per square inch}=(\frac{10^{-2}}{1550}\text{watts per square meter})$  

$\fbox{\begin\\\ 10^{-12}\text{watts per square inch}=6.4516\times10^{-6}\text{watts per square meter}\\\end{minispace}}$  

This implies that the value of $I$ is $6.4516\times10^{-6}$ watts per square meter.

Now, substitute the value of $I$ and $I_{0}$ in the equation $\fbox{\begin\\\ \beta(db)=10\text{log}(\frac{I}{I_{0}})\\\end{minispace}}$.

$\beta(db)=10\text{log}\left(\frac{6.4516\times10^_{-6}}{10^_{-12}}\right)$

$\beta(db)=10\text{log}(6451600)$

$\beta(db)=10\times6.809667$

$\fbox{\begin\\\ \beta(db)=68.09667433\\\end{minispace}}$

The approximated value of the above equation is as follows:

$\fbox{\begin\\\ \beta(db)=68.1\\\end{minispace}}$

From the above calculation it is concluded that the decibel level of the sound of a monster truck is $\fbox{\begin\\\ \beta(db)=68.1\\\end{minispace}}$.

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Answer details

Grade: Middle school

Subject: Mathematics  

Chapter: Logarithms

Keywords:  Logarithms, sound, intensity, power of sound, watts, decibel, decibel level, monster truck, logarithm function, unit, conversion of unit, square inch, square meter.