the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is…

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as $l = 10\\log\\frac{i}{i_0}$, where $i_0 = 10^{-12}$ and is the least intense sound a human ear can hear. jessica is listening to soft music at a sound intensity level of $10^{-9}$ on her computer while she does her homework. braylee is completing her homework while listening to very loud music at a sound intensity level of $10^{-3}$ on her headphones. how many times louder is braylees music than jessicas?\n\\(\\frac{1}{3}\\) times louder\n3 times louder\n30 times louder\n90 times louder

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as $l = 10\\log\\frac{i}{i_0}$, where $i_0 = 10^{-12}$ and is the least intense sound a human ear can hear. jessica is listening to soft music at a sound intensity level of $10^{-9}$ on her computer while she does her homework. braylee is completing her homework while listening to very loud music at a sound intensity level of $10^{-3}$ on her headphones. how many times louder is braylees music than jessicas?\n\\(\\frac{1}{3}\\) times louder\n3 times louder\n30 times louder\n90 times louder

Answer

Explanation:

Step1: Calculate loudness of Jessica's music

Let $I_{J}=10^{-9}$ be Jessica's sound - intensity. Using the formula $L = 10\log\frac{I}{I_{0}}$ with $I_{0}=10^{-12}$, we have $L_{J}=10\log\frac{10^{-9}}{10^{-12}}$. Since $\frac{10^{-9}}{10^{-12}}=10^{-9 - (- 12)}=10^{3}$, then $L_{J}=10\log(10^{3})$. And since $\log(10^{3}) = 3$, so $L_{J}=10\times3 = 30$ dB.

Step2: Calculate loudness of Braylee's music

Let $I_{B}=10^{-3}$ be Braylee's sound - intensity. Using the formula $L = 10\log\frac{I}{I_{0}}$ with $I_{0}=10^{-12}$, we have $L_{B}=10\log\frac{10^{-3}}{10^{-12}}$. Since $\frac{10^{-3}}{10^{-12}}=10^{-3-(-12)} = 10^{9}$, then $L_{B}=10\log(10^{9})$. And since $\log(10^{9}) = 9$, so $L_{B}=10\times9=90$ dB.

Step3: Find the ratio of loudness

To find how many times louder Braylee's music is than Jessica's, we calculate the ratio $\frac{L_{B}}{L_{J}}=\frac{90}{30}=3$.

Answer:

3 times louder