Two quick pre-calc questions

Question

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

Answer 1

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

Answer 2

"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.

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"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.