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 Jessica's loudness

Substitute (I = 10^{-9}) and (I_0=10^{-12}) into (L = 10\log\frac{I}{I_0}). [ \begin{align*} L_J&=10\log\frac{10^{-9}}{10^{-12}}\ &=10\log(10^{-9 + 12})\ &=10\log(10^{3})\ &=10\times3\ & = 30 \end{align*} ]

Step2: Calculate Braylee's loudness

Substitute (I = 10^{-3}) and (I_0 = 10^{-12}) into (L = 10\log\frac{I}{I_0}). [ \begin{align*} L_B&=10\log\frac{10^{-3}}{10^{-12}}\ &=10\log(10^{-3+12})\ &=10\log(10^{9})\ &=10\times9\ &=90 \end{align*} ]

Step3: Find the ratio of loudness

Calculate (\frac{L_B}{L_J}). [ \frac{L_B}{L_J}=\frac{90}{30}= 3 ]

Answer:

3 times louder.