The intensity level of sound A is 5.0dB greater than that of sound B and 3.0dB less than that of C. Determine the ratio(Ic/Ib) of the intensity of sound C to that of B.
To determine the ratio of the intensity of sound C (Ic) to that of B (Ib), we need to compare the decibel (dB) values and convert them back to intensity.
Let's assign the intensity level of sound B as Ib. Since sound A is 5.0 dB greater than sound B, the intensity level of sound A (Ia) can be expressed as:
Ia = Ib + 5.0 dB
Similarly, the intensity level of sound C (Ic) is 3.0 dB greater than sound A:
Ic = Ia + 3.0 dB
Now, to find the ratio Ic/Ib, we can divide the equation for Ic by the equation for Ib:
(Ic/Ib) = (Ia + 3.0 dB) / (Ib + 5.0 dB)
To proceed further, we need to understand the relationship between intensity and decibels in logarithmic units:
dB = 10 * log10(I1/I0)
Where I0 is the reference intensity (usually taken as the threshold of human hearing, which is 10^(-12) W/m²).
To convert the above equation into intensity units, we rewrite it as:
I1/I0 = 10^(dB/10)
Now let's substitute the values:
(Ic/Ib) = (10^((Ia+3.0 dB)/10)) / (10^((Ib+5.0 dB)/10))
Simplifying the equation, we can use the property of logarithms where 10^(a+b) = 10^a * 10^b:
(Ic/Ib) = (10^((Ia+3.0 dB)/10)) * (10^((-5.0 dB)/10)) / (10^((Ib)/10))
Next, cancel out similar terms in the numerator and denominator:
(Ic/Ib) = 10^(((Ia+3.0 dB)/10) - (Ib/10) - (5.0 dB/10))
Finally, we can substitute the values of Ia, Ib, and Ic from the earlier equations:
(Ic/Ib) = 10^(((Ib + 5.0 dB)/10 + 1/3) - (Ib/10) - (5.0 dB/10))
Now, we can calculate the ratio by evaluating the expression.