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 = 10log\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 = 10log\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 Braylee's loudness

Use the formula (L = 10\log\frac{I}{I_{0}}). For Braylee, (I = 10^{-3}) and (I_{0}=10^{-12}). [ \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*} ]

Step2: Calculate Jessica's loudness

For Jessica, (I = 10^{-9}) and (I_{0}=10^{-12}). [ \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*} ]

Step3: Find the ratio

Divide Braylee's loudness by Jessica's loudness: (\frac{L_{B}}{L_{J}}=\frac{90}{30}=3)

Answer:

3 times louder