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?\no $\\frac{1}{3}$ times louder\no 3 times louder\no 30 times louder\no 90 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?\no $\\frac{1}{3}$ times louder\no 3 times louder\no 30 times louder\no 90 times louder

Answer

Explanation:

Step1: Calculate Braylee's loudness

Let $I_1 = 10^{-3}$. Using the formula $L = 10\log\frac{I}{I_0}$ with $I_0=10^{-12}$, we have $L_1 = 10\log\frac{10^{-3}}{10^{-12}}=10\log(10^{-3 + 12})=10\log(10^{9})$. Since $\log(10^{9}) = 9$, then $L_1=10\times9 = 90$ decibels.

Step2: Calculate Jessica's loudness

Let $I_2 = 10^{-9}$. Using the formula $L = 10\log\frac{I}{I_0}$ with $I_0 = 10^{-12}$, we have $L_2=10\log\frac{10^{-9}}{10^{-12}}=10\log(10^{-9 + 12})=10\log(10^{3})$. Since $\log(10^{3}) = 3$, then $L_2 = 10\times3=30$ decibels.

Step3: Find the ratio of loudness

To find how many times louder Braylee's music is than Jessica's, we calculate $\frac{L_1}{L_2}=\frac{90}{30}=3$.

Answer:

3 times louder