Question

the loudness, l, measured in decibels (db), of a sound intensity, i, measured in watts per square meter, is defined as $l = 10\log\frac{i}{i_0}$, where $i_0 = 10^{-12}$ and is the least intense sound a human ear can hear. brandon is trying to take a nap, and he can barely hear his neighbor mowing the lawn. the sound intensity level that brandon can hear is $10^{-10}$. ahmad, brandons neighbor that lives across the street, is mowing the lawn, and the sound intensity level of the mower is $10^{-4}$. how does brandons sound intensity level compare to ahmads mower?
brandons sound intensity is $\frac{1}{4}$ the level of ahmads mower.
brandons sound intensity is $\frac{1}{6}$ the level of ahmads mower.
brandons sound intensity is 20 times the level of ahmads mower.
brandons sound intensity is 80 times the level of ahmads mower.

Explanation:

Step1: Set up ratio

We want to find the ratio of Brandon's sound - intensity $I_{B}$ to Ahmad's sound - intensity $I_{A}$. Given $I_{B}=10^{- 10}$ and $I_{A}=10^{-4}$. The ratio is $\frac{I_{B}}{I_{A}}$.

Step2: Calculate the ratio

Substitute the values into the ratio: $\frac{I_{B}}{I_{A}}=\frac{10^{-10}}{10^{-4}}$.
Using the rule of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$, we have $\frac{10^{-10}}{10^{-4}}=10^{-10-(-4)}=10^{-10 + 4}=10^{-6}=\frac{1}{10^{6}}$.

Answer:

Brandon's sound intensity is $\frac{1}{10^{6}}$ the level of Ahmad's mower. Since this option is not given, there might be a mis - understanding in the problem setup. If we consider the ratio the other way around $\frac{I_{A}}{I_{B}}=\frac{10^{-4}}{10^{-10}}=10^{-4-(-10)}=10^{6}$. But if we assume the question is asking for $\frac{I_{B}}{I_{A}}$, and we re - check the work, we note that if we want to express Brandon's intensity in terms of Ahmad's, we calculate $\frac{10^{-10}}{10^{-4}}=\frac{1}{10^{-4 + 10}}=\frac{1}{10^{6}}$. If we assume there is a typo in the options and we consider the closest conceptually, we may have misinterpreted the question as asking for $\frac{I_{A}}{I_{B}}$ which is $10^{6}$. If we go by the options and assume we made no error in the above steps and there is an error in options, we can say that none of the given options are correct. If we assume we should find $\frac{I_{B}}{I_{A}}$ and round or approximate in a non - standard way, we note that $\frac{10^{-10}}{10^{-4}}=\frac{1}{10^{6}}$ and among the options, the closest in terms of the concept of ratio is that Brandon's sound intensity is $\frac{1}{10^{6}}$ times Ahmad's, but if we have to pick from the given options, none of them are correct. If we assume we should find $\frac{I_{A}}{I_{B}}$ (which may be a mis - understanding of the question as stated), $\frac{10^{-4}}{10^{-10}} = 10^{6}$. But strictly based on the options and correct ratio calculation of $\frac{I_{B}}{I_{A}}$, no correct option.

If we assume the question is asking for how many times Ahmad's intensity is of Brandon's (opposite of what the question seems to ask), $\frac{I_{A}}{I_{B}}=\frac{10^{-4}}{10^{-10}}=10^{6}$ which is not among the options. If we assume we made a wrong start and we should find the ratio such that we get an option, we note that if we consider $\frac{I_{A}}{I_{B}}$ and approximate, we still don't get any of the options. So, based on the correct ratio calculation of Brandon's intensity to Ahmad's intensity $\frac{I_{B}}{I_{A}}=\frac{10^{-10}}{10^{-4}}=\frac{1}{10^{6}}$, there is no correct option among the given ones.