A 60 year old person has a threshold of hearing of 81.0 dB for a sound with frequency f=10,000 Hz. By what factor must the intensity of a sound wave of that frequency, audible to a typical young adult, (sound level=43.0 dB) be increased so that it is heard by the older person.

Question

A 60 year old person has a threshold of hearing of 81.0 dB for a sound with frequency f=10,000 Hz. By what factor must the intensity of a sound wave of that frequency, audible to a typical young adult, (sound level=43.0 dB) be increased so that it is heard by the older person.

Thanks for any and all help!

Answer 1

Hmmm. 81 - 43 is 38 db.

intentity = antilog (db level/10)
= antilog 3.8= 10^3.8

that is almost 7000 times.

Answer 2

Thank You so much Bob Pursley!!

Answer 3

To solve this question, we need to understand the relationship between sound intensity and sound level in decibels (dB).

The equation relating sound intensity (I) and sound level (L) is:

L = 10 * log10(I / I0)

Where I0 is the reference sound intensity (typically set at 10^-12 W/m^2).

First, we need to find the sound intensity (I1) for the sound wave audible to a typical young adult with a sound level of 43.0 dB:

43.0 = 10 * log10(I1 / 10^-12)

Dividing both sides of the equation by 10:

43.0 / 10 = log10(I1 / 10^-12)

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

4.30 = log10(I1) - log10(10^-12)

We know that log10(10^-12) is equal to -12, therefore:

4.30 = log10(I1) + 12

Now, let's calculate I1:

I1 = 10^(4.30 - 12)
I1 = 10^(-7.70)

Now, we need to find the sound intensity (I2) that the older person can hear with a threshold of hearing at 81.0 dB for a sound wave with a frequency of 10,000 Hz:

81.0 = 10 * log10(I2 / 10^-12)

Divide both sides of the equation by 10:

81.0 / 10 = log10(I2 / 10^-12)

Using the logarithmic property:

8.10 = log10(I2) - log10(10^-12)

We know that log10(10^-12) is equal to -12, therefore:

8.10 = log10(I2) + 12

Now, let's calculate I2:

I2 = 10^(8.10 - 12)
I2 = 10^(-3.90)

Finally, we can find the factor by which the sound intensity needs to be increased to be heard by the older person:

Factor = I2 / I1
Factor = (10^(-3.90)) / (10^(-7.70))

Using the division property of exponents (a^m / a^n = a^(m-n)):

Factor = 10^(-3.90 - (-7.70))
Factor = 10^(3.80)

Rounding to two decimal places:

Factor ≈ 630.96

Therefore, the intensity of the sound wave needs to be increased by a factor of approximately 630.96 for it to be heard by the older person.