5. The intensity of a sound is measured in watts per square meter $$ \left(\frac{\mathrm{W}}{\mathrm{m}^{2}}\right) $$, which represents how much power (measured in watts) the sound is exerting on a given area (measured in square meters). The quietest sound the average human ear can hear, $$ I_{0} $$, is known to measure $$ 10^{-12} \frac{\mathrm{~W}}{\mathrm{~m}^{2}} $$. Determine the decibel level of a sound with an intensity of $$ 10^{-3} \frac{\mathrm{~W}}{\mathrm{~m}^{2}} $$. Would a sound with an intensity of $$ 10^{-3} \frac{\mathrm{~W}}{\mathrm{~m}^{2}} $$ be considered unsafe?

Question

UpStudy AI · Accepted Answer

To determine the decibel (dB) level of a sound with a given intensity and assess its safety, we can follow these steps:

### 1. **Calculating the Decibel Level**

The decibel level ($$ \beta $$) of a sound is calculated using the formula:

$$
\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)
$$

Where:
- $$ I $$ is the intensity of the sound.
- $$ I_0 $$ is the reference intensity, which is the threshold of human hearing ($$ 10^{-12} \, \text{W/m}^2 $$).

Given:
- $$ I = 10^{-3} \, \text{W/m}^2 $$
- $$ I_0 = 10^{-12} \, \text{W/m}^2 $$

Plugging in the values:

$$
\beta = 10 \log_{10}\left(\frac{10^{-3}}{10^{-12}}\right) = 10 \log_{10}(10^{9}) = 10 \times 9 = 90 \, \text{dB}
$$

### 2. **Assessing Safety**

The safety of sound levels is generally evaluated based on their potential to cause hearing damage:

- **Safe Exposure:** Sounds below **85 dB** are typically considered safe for prolonged exposure.
- **Unsafe Exposure:** Sounds **above 85 dB** can pose a risk of hearing damage, especially with extended or repeated exposure.

Given that the calculated decibel level is **90 dB**, this exceeds the 85 dB threshold. Therefore:

- **A sound with an intensity of $$ 10^{-3} \, \text{W/m}^2 $$ (90 dB) is considered **unsafe** and can potentially cause hearing damage if exposure is prolonged.**

### **Summary**

- **Decibel Level:** 90 dB
- **Safety Assessment:** The sound is **unsafe** for prolonged exposure.

---

**Note:** It's essential to protect your hearing by avoiding prolonged exposure to high-decibel environments and using appropriate hearing protection when necessary.

**Answer Summary:**

A sound with intensity 10⁻³ W/m² has a decibel level of 90 dB and is considered unsafe for hearing.
A sound with an intensity of $$ 10^{-3} \, \text{W/m}^2 $$ has a decibel level of 90 dB and is considered unsafe for hearing.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Physics Questions