modeling with mathematics the blue whale can produce sound with an intensity that is 1 million times greater…

modeling with mathematics the blue whale can produce sound with an intensity that is 1 million times greater than a human can make. find the difference in the loudness of the sounds made by a blue whale and a human. for a sound with intensity i (in watts per square meter), the loudness l(i) of the sound (in decibels) is given by the function. intensity of a barely audible sound (about 10^(-12) watt per square meter). the difference in the loudness of the sounds made by a blue whale and a human decibels.

modeling with mathematics the blue whale can produce sound with an intensity that is 1 million times greater than a human can make. find the difference in the loudness of the sounds made by a blue whale and a human. for a sound with intensity i (in watts per square meter), the loudness l(i) of the sound (in decibels) is given by the function. intensity of a barely audible sound (about 10^(-12) watt per square meter). the difference in the loudness of the sounds made by a blue whale and a human decibels.

Answer

Explanation:

Step1: Recall the loud - ness formula

The loudness formula is $L(I)=10\log\left(\frac{I}{I_0}\right)$, where $I_0 = 10^{-12}\text{ W/m}^2$. Let the intensity of the human sound be $I_h$ and the intensity of the blue - whale sound be $I_w$. Given that $I_w = 1000000I_h=10^{6}I_h$.

Step2: Calculate the loudness of the human sound

$L_h = 10\log\left(\frac{I_h}{I_0}\right)$.

Step3: Calculate the loudness of the blue - whale sound

$L_w=10\log\left(\frac{I_w}{I_0}\right)=10\log\left(\frac{10^{6}I_h}{I_0}\right)$. Using the property of logarithms $\log(ab)=\log(a)+\log(b)$, we have $L_w = 10\left(\log(10^{6})+\log\left(\frac{I_h}{I_0}\right)\right)=10\times6 + 10\log\left(\frac{I_h}{I_0}\right)$.

Step4: Find the difference in loudness

$\Delta L=L_w - L_h$. Substitute the expressions for $L_w$ and $L_h$: $\Delta L=(60 + 10\log\left(\frac{I_h}{I_0}\right))-10\log\left(\frac{I_h}{I_0}\right)$. $60+10\log\left(\frac{I_h}{I_0}\right)-10\log\left(\frac{I_h}{I_0}\right)=60$.

Answer:

60