If the intensity of a sound wave is 10 W/m² at a distance of 5 m from the source, calculate the intensity at a distance of 20 m from the source.

1. Break Down the Problem

We need to determine the intensity of a sound wave at a distance of 20 m, given its intensity at 5 m is 10 W/m². This involves understanding how sound intensity varies with distance.

2. Relevant Concepts

Sound intensity follows the inverse square law, which states: $I_1 \cdot r_1^2 = I_2 \cdot r_2^2$ Where:

• $I_1$ = intensity at distance $r_1$
• $I_2$ = intensity at distance $r_2$

3. Analysis and Detail

Given:

• $I_1 = 10 \, \text{W/m}^2$
• $r_1 = 5 \, \text{m}$
• $r_2 = 20 \, \text{m}$

We can rearrange the formula to find $I_2$: $I_2 = \frac{I_1 \cdot r_1^2}{r_2^2}$ Substituting the known values: $I_2 = \frac{10 \, \text{W/m}^2 \cdot (5 \, \text{m})^2}{(20 \, \text{m})^2}$

Calculating further: $I_2 = \frac{10 \cdot 25}{400} = \frac{250}{400} = 0.625 \, \text{W/m}^2$

4. Verify and Summarize

We have verified the calculations, ensuring that all substitutions and simplifications were performed correctly. The intensity of the sound wave at a distance of 20 m from the source is $0.625 \, \text{W/m}^2$.

Final Answer

The intensity at a distance of 20 m from the source is $0.625 \, \text{W/m}^2$.