Math Problem Statement
The intensity of the sound of a certain television advertisement is 10 times greater than the intensity of the television program. By how many decibels does the loudness increase? Use the function $L\left(I\right)=10\log\frac{I}{I_0}$ . The loudness increases by decibels.
Solution
We are given that the intensity of the sound of the television advertisement is 10 times greater than the intensity of the television program. We need to calculate the increase in decibels using the given loudness function:
Where:
- is the intensity of the sound.
- is the reference intensity (a constant).
Step-by-Step Solution:
Let represent the intensity of the advertisement sound, and represent the intensity of the program sound. From the problem, we know:
We want to find the difference in loudness between the advertisement and the program.
The loudness of the advertisement and the program can be written as:
Step 1: Find the difference in loudness
The difference in loudness is given by:
Substitute the expressions for and :
Use the logarithmic property :
Step 2: Substitute
Substitute into the equation:
Simplify the fraction:
Step 3: Calculate the value of
Since , we get:
Final Answer:
The loudness increases by 10 decibels.
Would you like further explanation or more details on any step?
Here are 5 related questions you can explore:
- How does the loudness change if the intensity is 100 times greater instead of 10 times?
- What happens to the loudness if the intensity decreases instead of increases?
- How would you calculate the loudness change if the sound intensity ratio was 1000:1?
- What is the meaning of the reference intensity in the formula for loudness?
- How does the loudness relate to the physical energy of the sound waves?
Tip: In decibel calculations, the intensity ratio is often used to quantify loudness because the human ear perceives sound logarithmically, meaning a small increase in intensity can result in a large perceived increase in loudness.
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Math Problem Analysis
Mathematical Concepts
Logarithms
Sound Intensity
Decibels
Logarithmic Functions
Formulas
L(I) = 10 \log \left( \frac{I}{I_0} \right)
Delta L = 10 \log \left( \frac{I_{ad}}{I_{prog}} \right)
Theorems
Logarithmic Properties (log a - log b = log (a/b))
Suitable Grade Level
Grades 9-12