the loudness L of a sound in decibels is given L=10log10R, where R is the sound's relative intensity. if the intensity of a certain sound is tripled, by how many decibels does the sound increase?
due tomorrow, tuesday, april 5th, 2011.
help pleaseeeee ;)
say original R = 1
final R = 3
L1 = 10 log 10
L2 = 10 log 30
L2 - L1 = 10 (log 30-log 10)
= 10 log 30/10
=10 log 3
= 4.77 dB
ahhhhhh thank you! you're my mathematical hero :) hahaha
Well, tripling the intensity of a sound means multiplying it by 3. Now let's calculate the increase in decibels using the given formula:
L = 10log10R
Let's assume the initial intensity is R1, and the tripled intensity is R2. Therefore, R2 = 3R1.
To find the increase in decibels, we need to calculate the difference between L2 (sound with tripled intensity) and L1 (initial sound).
L2 = 10log10R2 = 10log10(3R1)
L1 = 10log10R1
Now, let's calculate the difference:
ΔL = L2 - L1 = 10log10(3R1) - 10log10R1
Using logarithmic properties, we can simplify this equation:
ΔL = 10(log103R1 - log10R1)
Now, the subtraction inside the parentheses can be simplified further:
ΔL = 10(log103R1/R1)
Since log103R1/R1 is equal to log103, we can simplify the equation even more:
ΔL = 10 * log103
Now, we can calculate the value of ΔL:
ΔL ≈ 10 * 0.477 = 4.77 decibels
So, the sound increases by approximately 4.77 decibels when its intensity is tripled.
I hope this helps, and best of luck with your assignment! If you need any more assistance, feel free to ask.
To find out by how many decibels the sound increases when its intensity is tripled, we can use the equation provided: L = 10log10(R).
Let's consider the original intensity as R1, and the new intensity after it is tripled as R2.
According to the equation, the original loudness is L1 = 10log10(R1), and the new loudness is L2 = 10log10(R2).
Given that the intensity is tripled, we can write R2 = 3R1.
Substituting this into the equation for loudness, we get L2 = 10log10(3R1).
Now, we can simplify this equation further using logarithm properties. Since log10(3R1) = log10(3) + log10(R1), the equation becomes L2 = 10(log10(3) + log10(R1)).
Using the property that log10(a) + log10(b) = log10(ab), we can simplify further:
L2 = 10(log10(3R1)) = 10(log10(3) + log10(R1)) = 10log10(3) + 10log10(R1).
Thus, the difference in loudness (increase) is L2 - L1.
Now, substitute L1 = 10log10(R1) and L2 = 10log10(3) + 10log10(R1) into the equation:
Difference in loudness = (10log10(3) + 10log10(R1)) - 10log10(R1)
= 10log10(3)
Using a calculator, log10(3) is approximately 0.4771.
Therefore, the sound increases in loudness by approximately 10 * 0.4771 decibels, which is approximately 4.77 decibels.
Thus, when the sound's intensity is tripled, it increases in loudness by approximately 4.77 decibels.
To find out by how many decibels the sound increases when its intensity is tripled, we can use the formula for loudness:
L = 10 * log10(R)
Let's assume the initial intensity is R1, and the final intensity after tripling is R2. We need to find the difference between the two loudness values, L1 and L2.
Given that the intensity is tripled, we can write:
R2 = 3 * R1
Now, let's substitute this value of R2 into the formula and solve for L2:
L2 = 10 * log10(3 * R1)
Using the logarithmic property log(a*b) = log(a) + log(b), we can simplify:
L2 = 10 * (log10(3) + log10(R1))
Using a calculator, find log10(3) ≈ 0.4771. Substitute this value into the equation:
L2 ≈ 10 * (0.4771 + log10(R1))
Now, let's assume the initial loudness is L1. We have:
L1 = 10 * log10(R1)
Now, we can find the difference between L2 and L1:
L2 - L1 ≈ 10 * (0.4771 + log10(R1)) - 10 * log10(R1)
Using logarithmic property log(a/b) = log(a) - log(b), we can simplify further:
L2 - L1 ≈ 10 * (0.4771 + log10(R1) - log10(R1))
Simplifying:
L2 - L1 ≈ 10 * 0.4771
Using a calculator, calculate 10 * 0.4771:
L2 - L1 ≈ 4.771
Therefore, the sound increases by approximately 4.771 decibels when its intensity is tripled.