SOLUTION: A law of physics states that the intensity of sound is inversely proportional to the square of the distance d of the source I: k/d^2 a. Use this model and the equation B=10log(I

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: A law of physics states that the intensity of sound is inversely proportional to the square of the distance d of the source I: k/d^2 a. Use this model and the equation B=10log(I      Log On


   

Question 815145: A law of physics states that the intensity of sound is inversely proportional to the square of the distance d of the source I: k/d^2
a. Use this model and the equation B=10log(I/I0) to show that the decibel levels B1 and B2 at distances d1 and d2 from a sound source are related by the equation B1= B2-20log(d1/d2)
I'm a little confused. Help?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since B=10log%28%28I%2FI%5B0%5D%29%29 then
B%5B1%5D=10log%28%28I%5B1%5D%2FI%5B0%5D%29%29 and B%5B2%5D=10log%28%28I%5B2%5D%2FI%5B0%5D%29%29

We want B%5B1%5D and B%5B2%5D in the same equation. So we will combine the two above by adding or subtracting them. I can see ahead that subtraction will make things a little easier so we will subtract the second equation from the first one:
B%5B1%5D-B%5B2%5D=10log%28%28I%5B1%5D%2FI%5B0%5D%29%29-10log%28%28I%5B2%5D%2FI%5B0%5D%29%29
Adding B%5B2%5D to each side:

As you can, the desired equation is starting to appear. We just need to manipulate the right side so that
  • the I's are gone
  • The d's are there
  • And it looks like the desired equation. The I's and d's are related by the equation: I+=+k%2Fd%5E2. So I%5B1%5D+=+k%2Fd%5B1%5D%5E2 and I%5B2%5D+=+k%2Fd%5B2%5D%5E2. Substituting for I%5B1%5D and I%5B2%5D we get:

    Most of the I's are gone and we have the d's. All that is left is to eliminate the I%5B0%5D's and the k's and make the equation look like the one we want. We do want only one log so I combine the two we have next. Factoring out 10:

    Now we can use the log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29 property to combine the logs:

    Rewriting the division of fractions as a multiplication by the reciprocal:

    ...and the I%5B0%5D's and the k's cancel out!

    which simplifies to:
    B%5B1%5D=B%5B2%5D%2B10%28log%28%28d%5B2%5D%5E2%2Fd%5B1%5D%5E2%29%29%29
    We're getting very close. Rewriting the fraction:
    B%5B1%5D=B%5B2%5D%2B10%28log%28%28%28d%5B2%5D%2Fd%5B1%5D%29%5E2%29%29%29
    Using the log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29 property:
    B%5B1%5D=B%5B2%5D%2B10%282log%28%28d%5B2%5D%2Fd%5B1%5D%29%29%29
    Multiplying the 10 and 2:
    B%5B1%5D=B%5B2%5D%2B20log%28%28d%5B2%5D%2Fd%5B1%5D%29%29