Question

sound intensity model: $l = 10logleft(\frac{i}{i_0}
ight)$
$l =$ loudness, in decibels (db); $i =$ sound intensity, in watts/m²; $i_0 = 10^{-12}$ watts/m²
the loudness of a jack hammer is 96 db. its sound intensity is about
the loudness of a compactor is 94 db. its sound intensity is about
the sound intensity of the jack hammer is about times the sound intensity of the compactor.
the loudness of a pile driver is 112 db. about how many times the sound intensity of the jackhammer is the sound intensity of a pile driver? round to the nearest ten.

Explanation:

Step1: Solve for sound - intensity formula

Given $L = 10\log(\frac{I}{I_0})$, we can rewrite it to solve for $I$. First, divide both sides by 10: $\frac{L}{10}=\log(\frac{I}{I_0})$. Then, use the inverse - logarithm property. Since $\log_{10}x = y$ is equivalent to $x = 10^y$, we have $\frac{I}{I_0}=10^{\frac{L}{10}}$, and $I = I_0\times10^{\frac{L}{10}}$.

Step2: Find sound intensity of jack - hammer

Given $L = 96$ dB and $I_0 = 10^{-12}$ watts/m². Substitute into the formula $I = I_0\times10^{\frac{L}{10}}$. So $I_{jack - hammer}=10^{-12}\times10^{\frac{96}{10}}=10^{-12}\times10^{9.6}=10^{-12 + 9.6}=10^{-2.4}\approx3.98\times10^{-3}$ watts/m².

Step3: Find sound intensity of compactor

Given $L = 94$ dB. Substitute into the formula $I = I_0\times10^{\frac{L}{10}}$. So $I_{compactor}=10^{-12}\times10^{\frac{94}{10}}=10^{-12}\times10^{9.4}=10^{-12 + 9.4}=10^{-2.6}\approx2.51\times10^{-3}$ watts/m².

Step4: Find ratio of jack - hammer to compactor sound intensity

The ratio $\frac{I_{jack - hammer}}{I_{compactor}}=\frac{10^{-2.4}}{10^{-2.6}}=10^{-2.4+2.6}=10^{0.2}\approx1.58$.

Step5: Find sound intensity of pile - driver

Given $L = 112$ dB. Substitute into the formula $I = I_0\times10^{\frac{L}{10}}$. So $I_{pile - driver}=10^{-12}\times10^{\frac{112}{10}}=10^{-12}\times10^{11.2}=10^{-12 + 11.2}=10^{-0.8}\approx0.158$ watts/m².

Step6: Find ratio of pile - driver to jack - hammer sound intensity

The ratio $\frac{I_{pile - driver}}{I_{jack - hammer}}=\frac{10^{-0.8}}{10^{-2.4}}=10^{-0.8 + 2.4}=10^{1.6}\approx39.8\approx40$.

Answer:

The sound intensity of the jack - hammer is about $3.98\times10^{-3}$ watts/m².
The sound intensity of the compactor is about $2.51\times10^{-3}$ watts/m².
The sound intensity of the jack - hammer is about $1.6$ times the sound intensity of the compactor.
The sound intensity of the pile - driver is about 40 times the sound intensity of the jack - hammer.