o quantify the intensity of​ sound, the decibel scale was developed. The formula for loudness L on the decibel scale is Upper L equals 10 log left parenthesis StartFraction Upper I Over Upper I 0 EndFraction right parenthesis

Question

​, where Upper I 0 is the intensity of sound just below the threshold of​ hearing, which is approximately 10 Superscript negative 16 watt per square centimeter. Find the decibel reading for a sound with intensity 23 comma 000 times Upper I 0.
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Part 1
The decibel reading is approximately
  
enter your response here. ​(Round to the nearest whole number as​ needed.)

Answer 1

To find the decibel reading for a sound with intensity \( I = 23,000 \times I_0 \), we can use the formula:

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

Since \( I = 23,000 \times I_0 \), we can substitute this into the formula:

\[ L = 10 \log\left(\frac{23,000 \times I_0}{I_0}\right) \]

This simplifies to:

\[ L = 10 \log(23,000) \]

Next, we can calculate \( \log(23,000) \).

Using a calculator:

\[ \log(23,000) \approx 4.362 \]

Now plug this value back into the formula:

\[ L \approx 10 \times 4.362 = 43.62 \]

Rounding to the nearest whole number, we get:

\[ L \approx 44 \]

Thus, the decibel reading is approximately 44.