The loudness​ L(x), measured in​ decibels, of a sound of intensity​ x, measured in watts per square​ meter, is defined as Upper L left parenthesis x right parenthesis equals 10 log StartFraction x Over Upper I 0 EndFraction comma
 where Upper I 0 equals 10 Superscript negative 12 watt per square meter is the least intense sound that a human ear can detect.  Determine the​ loudness, in​ decibels, of a sound with intensity of x equals 10 Superscript negative 9.4 watt per square meter.

The formula for the loudness \( L(x) \), measured in decibels (dB), is given by:

\[
L(x) = 10 \log \left( \frac{x}{I_0} \right)
\]

where \( I_0 = 10^{-12} \, \text{watt/m}^2 \) is the least intense sound a human ear can detect, and \( x \) is the intensity of the sound in watts per square meter.

We are given:
- \( x = 10^{-9.4} \, \text{watt/m}^2 \)
- \( I_0 = 10^{-12} \, \text{watt/m}^2 \)

Substitute these values into the formula for \( L(x) \):

\[
L(x) = 10 \log \left( \frac{10^{-9.4}}{10^{-12}} \right)
\]

Simplify the argument inside the logarithm:

\[
\frac{10^{-9.4}}{10^{-12}} = 10^{-9.4 + 12} = 10^{2.6}
\]

Now the formula becomes:

\[
L(x) = 10 \log(10^{2.6})
\]

Since \( \log(10^{2.6}) = 2.6 \), we have:

\[
L(x) = 10 \times 2.6 = 26 \, \text{decibels}
\]

Thus, the loudness of the sound is **26 decibels**.

Would you like further clarification, or do you have any questions?

Here are 5 related questions for practice:
1. What is the loudness of a sound with intensity \( 10^{-8} \, \text{watt/m}^2 \)?
2. How does doubling the intensity of sound affect the decibel level?
3. If a sound has a loudness of 50 decibels, what is its intensity in watts per square meter?
4. What is the logarithmic relationship between sound intensity and loudness?
5. How does a 10 dB increase affect perceived loudness by the human ear?

**Tip:** Remember, a 10 dB increase roughly doubles the perceived loudness of a sound to the human ear!

The loudness L(x), measured in decibels, of a sound of intensity x, measured in watts per square meter, is defined as L(x) = 10 log (x / I_0), where I_0 = 10^(-12) watt per square meter is the least intense sound that a human ear can detect. Determine the loudness, in decibels, of a sound with intensity x = 10^(-9.4) watt per square meter.

Find out how to calculate the loudness of a sound using the formula L(x) = 10 log (x / I_0) where the sound intensity is 10^(-9.4) W/m². This step-by-step guide uses logarithmic properties to determine the sound's loudness in decibels, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation