The loudness L of sound in decibels is given by L= 10log (l/lo) where l is the intensity of sound and lo is the intensity of the least audible sound. If lo= 10^-12 w/m^2, about how many times more intense is a 108 decibel sound than a 100 decibel sound?
To compare the intensity of a 108 decibel sound to a 100 decibel sound, we need to calculate the difference in decibel levels and then convert it back to intensity.
The formula to calculate the difference in decibel levels is:
ΔL = L2 - L1
Where ΔL is the difference in decibels, L2 is the larger decibel level, and L1 is the smaller decibel level.
Let's calculate the difference in decibels:
ΔL = 108 dB - 100 dB
ΔL = 8 dB
The difference in decibels is 8 dB.
Now, we will convert this difference back to intensity using the formula given:
L = 10log(l/lo)
To find the intensity ratio, we need to raise 10 to the power of the difference in decibels (ΔL / 10):
Intensity ratio = 10^(ΔL/10)
Plugging in the values:
Intensity ratio = 10^(8/10)
Intensity ratio = 10^(0.8)
Using a calculator, we find that 10^(0.8) is approximately 6.31.
Therefore, a 108 decibel sound is about 6.31 times more intense than a 100 decibel sound.
To find out how many times more intense a 108 decibel sound is compared to a 100 decibel sound, we need to convert the decibel values back to intensities and then compare them.
Given:
Loudness of sound (L1) = 108 decibels
Loudness of sound (L2) = 100 decibels
Intensity of the least audible sound (lo) = 10^-12 W/m^2
We can use the formula for converting decibels to intensities:
L = 10 log (l/lo)
Rearranging the formula:
l/lo = 10^(L/10)
Now let's calculate the intensities (l1 and l2) for L1 and L2:
For L1:
l1/lo = 10^(L1/10) = 10^(108/10) = 10^10.8
For L2:
l2/lo = 10^(L2/10) = 10^(100/10) = 10^10
To find out how many times more intense L1 (108 decibels) is compared to L2 (100 decibels), divide l1 by l2:
(l1/lo) / (l2/lo) = l1/l2
Substituting the values:
l1/l2 = (10^10.8)/(10^10) = 10^(10.8-10) = 10^(0.8) ≈ 6.31
Therefore, a 108 decibel sound is approximately 6.31 times more intense than a 100 decibel sound.