Answer
The intensity level of a sound in decibels can be calculated using the formula:
IL = 10*log(I/Io)
where IL is the intensity level in decibels, I is the sound intensity, and Io is the reference intensity (1 x 10^-12 W/m^2). We can use this formula to calculate the intensity level of a 23-dB sound after being amplified by different factors:
(a) Ten thousand times:
The amplification factor is 10,000, which means the sound intensity will be increased by a factor of 10,000. Therefore, the new sound intensity is:
I_new = 10,000 * I_old
The new intensity level can be calculated as:
IL_new = 10log(I_new/Io) = 10log(10,000 * I_old/Io) = 10log(I_old/Io) + 10log(10,000) = IL_old + 40 dB
Therefore, the intensity level of the 23-dB sound after being amplified ten thousand times is:
IL_new = 23 + 40 = 63 dB
(b) A million times:
The amplification factor is 1,000,000, which means the sound intensity will be increased by a factor of 1,000,000. Therefore, the new sound intensity is:
I_new = 1,000,000 * I_old
The new intensity level can be calculated as:
IL_new = 10log(I_new/Io) = 10log(1,000,000 * I_old/Io) = 10log(I_old/Io) + 10log(1,000,000) = IL_old + 60 dB
Therefore, the intensity level of the 23-dB sound after being amplified a million times is:
IL_new = 23 + 60 = 83 dB
(c) A billion times:
The amplification factor is 1,000,000,000, which means the sound intensity will be increased by a factor of 1,000,000,000. Therefore, the new sound intensity is:
I_new = 1,000,000,000 * I_old
The new intensity level can be calculated as:
IL_new = 10log(I_new/Io) = 10log(1,000,000,000 * I_old/Io) = 10log(I_old/Io) + 10log(1,000,000,000) = IL_old + 90 dB
Therefore, the intensity level of the 23-dB sound after being amplified a billion times is:
IL_new = 23 + 90 = 113 dB
Work Step by Step
sound intensity level $= 10log\frac{I}{I_o}$
