Sound intensities, I, are often compared with the threshold of human hearing, I_0, which is about 10^-12 watts per meter squared. The relative intensity, R, of a sound is given by the equation: R = 10log(I/I_0)
i. The intensity of a whisper is about 300 times as loud as the threshold of human hearing. Find the relative intensity, R, of a whisper in decibels.
ii. Suppose a burglar alarm has a rating of 120 decibels. Compare this intensity to the threshold of human hearing.
i. what is 10log300? Ans 10*2*log3 work that out, about 9db
ii. Threshold human hearing: 0 db so 120/10=12, so ...
To find the relative intensity, R, of a sound in decibels using the given equation R = 10log(I/I_0), we need to have the value of the sound intensity, I.
i. The intensity of a whisper is said to be about 300 times as loud as the threshold of human hearing. Therefore, we can calculate the intensity of the whisper by multiplying the threshold intensity, I_0, by 300.
I = 300 * I_0
I = 300 * 10^-12 watts per meter squared
To work with the values in decibels, we can plug this value of I into the equation R = 10log(I/I_0) and calculate R.
R = 10log(300 * 10^-12 / 10^-12)
Simplifying the above equation:
R = 10log(300)
R = 10 * 2.477
Using a calculator, we find that R ≈ 24.77 decibels.
Therefore, the relative intensity of a whisper is approximately 24.77 decibels.
ii. To compare the intensity of a burglar alarm, given in decibels, to the threshold of human hearing, we need to find the sound intensity, I, corresponding to the given decibel rating of 120.
Using the equation R = 10log(I/I_0), we can rearrange it to solve for I:
R = 10log(I/I_0)
120 = 10log(I/10^-12)
Dividing both sides of the equation by 10:
12 = log(I/10^-12)
Converting to exponential form:
I/10^-12 = 10^12
Simplifying the equation:
I = 10^12 * 10^-12
I = 10^0
I = 1 watt per meter squared
Therefore, an intensity of 1 watt per meter squared corresponds to a decibel rating of 120.
Comparing this intensity to the threshold of human hearing, we can see that the burglar alarm is much louder. The threshold intensity, I_0, is about 10^-12 watts per meter squared, while the burglar alarm has an intensity of 1 watt per meter squared.
In conclusion, the burglar alarm is much more intense, approximately 1 trillion times louder than the threshold of human hearing.