(a) By what factor is one sound more intense than another if the sound has a level 17.0 dB higher than the other?

(b) If one sound has a level 32.0 dB less than another, what is the ratio of their intensities?

(a)

L=10 log₁₀ (I₁/I₀)
ΔL= L₁ - L₂ =
=10 log₁₀(I₁/I₀) – 10log₁₀(I₂/I₀) =
=10 log₁₀(I₁/I₂)
ΔL/10 = log₁₀(I₁/I₂)
I₁/I₂ =10^( ΔL/10) =10^1.7 =50.1
(b)
I₁/I₂ =10^( ΔL/10) =10^3.2 =1584.8
I₂/I₁= 6.3•10⁻⁴

To compute the factor by which one sound is more intense than another, given the difference in sound levels, we need to understand the relationship between sound levels and intensities.

(a) The relationship between sound levels and intensities is given by the formula:

L2 = L1 + 10 * log10(I2/I1),

where L1 and L2 are the sound levels in decibels (dB) for sound 1 and sound 2, and I1 and I2 are their corresponding intensities. In this case, we are given that the sound level of sound 2 is 17.0 dB higher than sound 1.

Let's denote L1 as the sound level for sound 1. Therefore, the sound level for sound 2 is L1 + 17.0 dB.

Plugging this information into the formula, we get:

L1 + 17.0 = L1 + 10 * log10(I2/I1).

Now, let's solve for the factor by which sound 2 is more intense than sound 1:

10 * log10(I2/I1) = 17.0.

Dividing both sides of the equation by 10, we get:

log10(I2/I1) = 1.7.

To convert this to a ratio, we raise both sides of the equation to the power of 10:

I2/I1 = 10^1.7.

Finally, we can compute the factor by which one sound is more intense than the other by solving for I2/I1:

I2/I1 = 50.12.

Therefore, one sound is approximately 50.12 times more intense than the other sound.

(b) Using the same formula as above, we can determine the ratio of intensities when one sound level is 32.0 dB less than the other.

Let's denote L1 as the sound level for sound 1. Therefore, the sound level for sound 2 is L1 - 32.0 dB.

Plugging this information into the formula, we get:

L1 - 32.0 = L1 + 10 * log10(I2/I1).

Simplifying the equation, we have:

-32.0 = 10 * log10(I2/I1).

Dividing both sides by 10, we get:

-3.2 = log10(I2/I1).

To convert this to a ratio, we raise both sides to the power of 10:

I2/I1 = 10^-3.2.

Finally, we can compute the ratio of intensities by solving for I2/I1:

I2/I1 = 0.000631.

Therefore, the ratio of intensities is approximately 0.000631.