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. what is the approximate loudness of a rock concert with a sound intensity of $10^{-1}$?

$\bigcirc$ 2 db
$\bigcirc$ 22 db
$\bigcirc$ 60 db
$\bigcirc$ 110 db

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. what is the approximate loudness of a rock concert with a sound intensity of $10^{-1}$?

$\bigcirc$ 2 db
$\bigcirc$ 22 db
$\bigcirc$ 60 db
$\bigcirc$ 110 db

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Laws of Exponents" ], "new_concepts": [ "Logarithmic Models", "Decibel Scale Calculations" ], "current_concepts": [ "Laws of Exponents", "Logarithmic Models", "Decibel Scale Calculations" ] } ## Identify the given values and formula $$ \begin{aligned} L &= 10 \log \frac{I}{I_0} \ I_0 &= 10^{-12} \ I &= 10^{-1} \end{aligned} $$ ## Substitute the values and simplify the fraction $$ \begin{aligned} \frac{I}{I_0} &= \frac{10^{-1}}{10^{-12}} \ &= 10^{-1 - (-12)} \ &= 10^{11} \end{aligned} $$ ## Calculate the loudness $$ \begin{aligned} L &= 10 \log(10^{11}) \ &= 10 \times 11 \ &= 110\text{ Db} \end{aligned} $$

2 Db

22 Db

60 Db

110 Db

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Logarithmic Models" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the given values and formula

Substitute the values and simplify the fraction

Calculate the loudness

Answer: