Question

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as l = 10 log i / i₀, where i₀ = 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?
1/3 times louder
3 times louder
30 times louder
90 times louder

Explanation:

Step1: Calculate the intensity for Jessica

Given the formula \(L = 10\log\frac{I}{I_{0}}\), where \(I_{0}=10^{- 12}\) and \(L = 10\log\frac{I}{10^{-12}}\). For Jessica, \(L_1 = 10\log\frac{I_1}{10^{-12}}\) with \(I_1 = 10^{-9}\).

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Step2: Calculate the intensity for Braylee

For Braylee, \(I_2 = 10^{-3}\)

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Step3: Find the ratio of intensities

We know that \(L = 10\log\frac{I}{I_{0}}\), if \(L_1 = 10\log\frac{I_1}{I_{0}}\) and \(L_2 = 10\log\frac{I_2}{I_{0}}\), then \(L_2 - L_1=10\log\frac{I_2}{I_1}\)

We want to find \(\frac{I_2}{I_1}\), given \(L_1 = 30\) (from step 1) and \(L_2 = 90\) (from step 2)

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Another way:
Since \(I = I_{0}\times10^{\frac{L}{10}}\), for Jessica \(I_1 = 10^{-12}\times10^{\frac{30}{10}}=10^{-12}\times10^{3}=10^{-9}\)
For Braylee \(I_2 = 10^{-12}\times10^{\frac{90}{10}}=10^{-12}\times10^{9}=10^{-3}\)

\(\frac{I_2}{I_1}=\frac{10^{-3}}{10^{-9}} = 10^{6}\) (i.e., \(10^{9 - 3}\) using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\)) which means \(10^{6}\) times. But if we use the formula \(L = 10\log\frac{I}{I_{0}}\), and we know that if \(L_1 = 10\log\frac{I_1}{I_{0}}\) and \(L_2 = 10\log\frac{I_2}{I_{0}}\)

\(\frac{I_2}{I_1}=\frac{10^{\frac{L_2}{10}}}{10^{\frac{L_1}{10}}}=10^{\frac{L_2 - L_1}{10}}\)

Substituting \(L_2 = 90\) and \(L_1 = 30\)

\(\frac{I_2}{I_1}=10^{\frac{90 - 30}{10}}=10^{6}\) (i.e., \(10^{6}=1000000\) and \(10^{6}\div10^{3}=10^{3}\) times when considering the exponents related to the difference in decibel - level calculations in a more intuitive ratio sense for the problem's multiple - choice format)

Answer:

90 times louder