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\n$\\frac{1}{3}$ times louder\n\n3 times louder\n\n30 times louder\n\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\n$\\frac{1}{3}$ times louder\n\n3 times louder\n\n30 times louder\n\n90 times louder

Answer

Answer:

C. 30 times louder

Explanation:

Step1: Calcular el nivel de decibeles de Jessica

Usamos la fórmula $L = 10\log\frac{I}{I_0}$. Dado que $I_{Jessica}=10^{-9}$ y $I_0 = 10^{-12}$, entonces $L_{Jessica}=10\log\frac{10^{-9}}{10^{-12}}=10\log(10^{-9 + 12})=10\log(10^{3}) = 10\times3=30$ dB.

Step2: Calcular el nivel de decibeles de Braylee

Dado que $I_{Braylee}=10^{-3}$ y $I_0 = 10^{-12}$, entonces $L_{Braylee}=10\log\frac{10^{-3}}{10^{-12}}=10\log(10^{-3+12})=10\log(10^{9}) = 10\times9 = 90$ dB.

Step3: Encontrar la diferencia

Para saber cuántas veces más alta es la música de Braylee que la de Jessica, restamos los niveles de decibeles: $\frac{L_{Braylee}-L_{Jessica}}{10}=\frac{90 - 30}{10}=6$. Pero esto no es la respuesta final. Lo que queremos es $\frac{10^{\frac{L_{Braylee}}{10}}}{10^{\frac{L_{Jessica}}{10}}}=10^{\frac{L_{Braylee}-L_{Jessica}}{10}}$. Sustituyendo los valores, $10^{\frac{90 - 30}{10}}=10^{6/1}= 10^{3}=30$ veces.