The loudness of sound (L, measured in decibels) perceived by the human ear depends on intensity levels (I) according to , where is the standard threshold of audibility. If then using the change-of-base formula, we get . At what rate is the loudness changing with respect to x when the intensity is 1,000 times the standard threshold of audibility (that is, when x = 1,000)?

Question

The loudness of sound (L, measured in decibels) perceived by the human ear depends on intensity levels (I) according to  , where  is the standard threshold of audibility. If  then using the change-of-base formula, we get  . At what rate is the loudness changing with respect to x when the intensity is 1,000 times the standard threshold of audibility (that is, when x = 1,000)? ​

Quizplus · Accepted Answer

Using the given formula, we have:

L = 10 log(I/I0)

Taking the derivative with respect to x, we get:

dL/dx = 10/ln(10) * 1/I * dI/dx

We are given that I = 1000I0, so:

dI/dx = 0 (since I0 is a constant)

Plugging in I = 1000I0 and I0 = 10^-12 W/m^2, we get:

L = 10 log(1000) = 30 dB

Now we can calculate:

dL/dx = 10/ln(10) * 1/(1000I0) * 0 = 0

Therefore, the rate of change of loudness with respect to x is 0 when the intensity is 1000 times the standard threshold of audibility.

However, we are asked for the rate of change at this point, so the answer is actually:

dL/dx = 0 * ln(10)/10 = 0 dB/dx

This is equivalent to choice B.

The Loudness of Sound (L, Measured in Decibels) Perceived by the Human