Let's solve the two problems step by step.

Question 17:

What is the sound intensity in watts per square meter ($W/m^2$) of an engine that is rated at 82 dB?

The formula to convert sound intensity level ($L$) in decibels (dB) to intensity ($I$) in $W/m^2$ is: $L = 10 \cdot \log_{10}\left(\frac{I}{I_0}\right)$ where:

• $L$ is the sound level in decibels,
• $I_0 = 10^{-12} \, W/m^2$ is the reference intensity (threshold of human hearing).

Rearranging for $I$: $I = I_0 \cdot 10^{\frac{L}{10}}$

Substituting $L = 82 \, \text{dB}$ and $I_0 = 10^{-12} \, W/m^2$: $I = 10^{-12} \cdot 10^{\frac{82}{10}}$ $I = 10^{-12} \cdot 10^{8.2}$ $I = 10^{-12 + 8.2} = 10^{-3.8} \, W/m^2$

Simplify further: $I \approx 1.58 \times 10^{-4} \, W/m^2$

Answer: The sound intensity is approximately $1.58 \times 10^{-4} \, W/m^2$.

Question 18:

How many times more intense is an earthquake of magnitude 6.2 than an earthquake of magnitude 5.5?

The Richter scale is logarithmic, and the intensity ratio of two earthquakes is given by: $\text{Intensity Ratio} = 10^{(M_1 - M_2)}$ where $M_1$ and $M_2$ are the magnitudes of the two earthquakes.

Substituting $M_1 = 6.2$ and $M_2 = 5.5$: $\text{Intensity Ratio} = 10^{(6.2 - 5.5)} = 10^{0.7}$

Using $10^{0.7} \approx 5.01$: $\text{Intensity Ratio} \approx 5.01$

Answer: The earthquake of magnitude 6.2 is approximately 5.01 times more intense than the earthquake of magnitude 5.5.

Would you like further explanations or details?

Here are 5 related questions for deeper understanding:

1. What does the decibel scale measure, and why is it logarithmic?
2. How is the reference intensity ($I_0$) chosen for sound measurements?
3. Why is the Richter scale logarithmic, and what does each unit represent in terms of energy release?
4. How would the sound intensity change for an engine rated at 90 dB?
5. What is the physical meaning of the sound intensity level being $1.58 \times 10^{-4} \, W/m^2$?

Tip: Logarithmic scales like decibels and Richter scales are used to handle very large variations in intensity or energy efficiently.