A construction explosion has an intensity I of 1.25 x 10–^4 W/m^2. Find the loudness of the sound in decibels if L=101og I/I0 and I 0 =10^-2 W/m^2. Round to the nearest tenth.
Given that the formula for loudness L in decibels is L = 10log(I/I0), where I is the intensity of the sound and I0 is the reference intensity, we are given:
I = 1.25 x 10^(-4) W/m^2
I0 = 10^(-2) W/m^2
Substituting these values into the formula, we have:
L = 10log(1.25 x 10^(-4) / 10^(-2))
Simplifying the fraction inside the logarithm, we get:
L = 10log(1.25 x 10^(-4) / 10^(-2))
= 10log(1.25 x 10^(-4) x 10^(2))
= 10log(1.25 x 10^(-2))
= 10log(1.25)
Using the logarithmic property of log(ab) = log(a) + log(b), we can rewrite it as:
L = 10(log(1.25) + log(10^(-2)))
= 10log(1.25) + 10log(10^(-2))
Since log(10) = 1, we have:
L = 10log(1.25) + 10(-2)
= 10log(1.25) - 20
Using a calculator, we find that log(1.25) ≈ 0.0969, rounding to four decimal places. Therefore:
L ≈ (10 x 0.0969) - 20
= 0.969 - 20
= -19.031
Rounding to the nearest tenth, the loudness of the sound in decibels is approximately -19.0 decibels.