### Correct Option: A. 4.00 ### Detailed Explanation: To understand how the intensity of sound changes with distance from a point source, we need to consider the relationship between intensity and distance. The intensity \( I \) of sound is defined as the power \( P \) per unit area \( A \): \[ I = \frac{P}{A} \] For a point source emitting sound uniformly in all directions, the sound spreads out over the surface of a sphere. The area \( A \) of a sphere is given by the formula: \[ A = 4\pi r^2 \] where \( r \) is the radius (or the distance from the source). 1. **Initial Intensity**: Let's denote the initial distance from the sound source as \( r_1 \). The intensity at this distance is: \[ I_1 = \frac{P}{4\pi r_1^2} \] 2. **New Intensity**: If we double the distance, the new distance becomes \( r_2 = 2r_1 \). The intensity at this new distance is: \[ I_2 = \frac{P}{4\pi (2r_1)^2} = \frac{P}{4\pi (4r_1^2)} = \frac{P}{16\pi r_1^2} \] 3. **Comparing Intensities**: Now, we can find the factor by which the intensity decreases when the distance is doubled. We can express the new intensity \( I_2 \) in terms of the initial intensity \( I_1 \): \[ I_2 = \frac{1}{4} I_1 \] This shows that when the distance from the point source is doubled, the intensity decreases to one-fourth of its original value. ### Why the Other Options are Wrong: - **Option B: 2.00**: This option suggests that the intensity would decrease by a factor of 2. However, as we derived, the intensity actually decreases by a factor of 4, not 2. This option misrepresents the relationship between intensity and distance. - **Option C: 0.50**: This option implies that the intensity would be half of the original intensity. This is incorrect because the relationship is quadratic (due to the area of the sphere), meaning that doubling the distance results in a decrease by a factor of 4, not just 2. - **Option D: 0.25**: While this option numerically represents the new intensity as a fraction of the original intensity, it does not answer the question correctly. The question asks for the factor by which the intensity decreases, which is 4. Therefore, while 0.25 is correct in terms of the new intensity, it does not reflect the decrease factor. ### Summary of Key Points: - The intensity of sound from a point source decreases with the square of the distance from the source. - Doubling the distance results in the intensity decreasing to one-fourth of its original value. - The correct factor by which intensity decreases when distance is doubled is 4.00. - Understanding the relationship between intensity and distance is crucial for solving problems related to sound and other wave phenomena.