Search
Ask a question
0 0

Two quick pre-calc questions

Sound A is 31 dB. Sound B is 62 dB. How many times more intense is Sound B than Sound A
 
How many times more powerful is an earthquake measuring 5.3 on the richter scale than an earthquake measuring 4.2 on the richter scale
Tutors, please sign in to answer this question.

2 Answers

The Decibel scale is logarithmic, with every 10 dB representing a tenfold increase in intensity.  Therefore sound intensity is proportional to 10^(x/10), where x is the Decibel measure of the sound.  It follows that the intensity ratio of sound B to sound A is:
 
I_B / I_A = 10^((62 - 31)/10) = 10^3.1 ≈ 1258.9
 
The Richter scale is also logarithmic, with every 1.0 on the scale representing a tenfold increase in intensity.  The ratio of the earthquake intensities is:
 
10^(5.3 - 4.2) ≈ 12.589
"Sound A is 31 dB. Sound B is 62 dB. How many times more intense is Sound B than Sound A?"

D dB := 10 log(I*10^12 watts/m^2)

31 dB = 10 log(I_A*10^12); 62 dB = 10 log(I_B*10^12)

3.1 = log(I_A*10^12); 6.2 = log(I_B*10^12)

Find Equivalent Exponential Forms:

10^3.1 = I_A*10^12; 10^6.2 = I_B*10^12

Divide equation with B by equation with A:

10^6.2/10^3.1 = I_B*10^12/(I_A*10^12) = I_B/I_A

I_B/I_A = 10^(6.2-3.1) = 10^3.1 ≈ 1258.925411794168

Sound B is exactly 1000*10^(1/10) times more intense than sound A,
or Sound B is approximately 1258.925411794168 times more intense than sound A.

=====

"How many times more powerful is an earthquake measuring 5.3 on the Richter Scale than an earthquake measuring 4.2 on the Richter Scale."

R := log(I/I_0); I is intensity, I_0 is very small reference intensity.

5.3 = log(I_A/I_0); 4.2 = log(I_B/I_0)

Find Equivalent Exponential Forms:

10^5.3 = I_A/I_0; 10^4.2 = I_B/I_0

Divide first equation by second:

10^5.3/10^4.2 = (I_A/I_0)/(I_B/I_0) = I_A/I_B

I_A/I_B = 10^(5.3-4.2)

I_A/I_B = 10^1.1 = 10*10^(1/10) exactly, or

I_A/I_B ≈ 12.58925411794167 times more powerful.