Question

The intensity at a distance of 6.0 m from a source that is radiating equally in all directions is 6.0 × 10-10 W/m2. What is the intensity level in dB? The threshold of human hearing is 1.0 × 10-12 W/m2.

Answer 1

To calculate the intensity level in decibels (dB), we can use the formula:

Intensity level (dB) = 10 * log10(I/I₀)

Where I is the intensity of the sound and I₀ is the reference intensity.

Given:

Intensity at a distance of 6.0 m = 6.0 × 10^(-10) W/m^2

Threshold of human hearing = 1.0 × 10^(-12) W/m^2

Substituting the values into the formula:

Intensity level (dB) = 10 * log10((6.0 × 10^(-10)) / (1.0 × 10^(-12)))

Simplifying:

Intensity level (dB) = 10 * log10(6.0 × 10^(-10) / 1.0 × 10^(-12))

Intensity level (dB) = 10 * log10(6.0 × 10^2)

Using the logarithmic property log10(a/b) = log10(a) - log10(b):

Intensity level (dB) = 10 * (log10(6.0) + log10(10^2))

Intensity level (dB) = 10 * (log10(6.0) + 2)

Using a calculator or logarithm table, we can find that log10(6.0) is approximately 0.7782:

Intensity level (dB) = 10 * (0.7782 + 2)

Intensity level (dB) = 10 * 2.7782

Intensity level (dB) ≈ 27.782 dB

Therefore, the intensity level at a distance of 6.0 m from the source is approximately 27.782 dB.