Question 1114137: After some complaints, it was found that the music from an outdoor concert was 5 decibels louder than the city's allowable level of 100 decibels. The loudness L of sound in decibels is given by L=10log(i/i base 0), where i is the intensity of sound and i base 0 is the intensity of the softest audible sound. How many times more intense is the concert sound than the allowable level?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! allowable level = 100 decibels.
actual level = 105 decibels.
L = 10 * log(i/i0)
i = intensity of sound.
i0 = intensity of the softest audible sound.
if we let k = i0, then the formula becomes L = 10 * log(i/k).
if we let c = intensity of the concert and we let a = allowable level of intensity, then the formula for the concert becomes 105 = 10 * log(c/k), and the formula for the allowable level becomes 100 = 10 * log(a/k).
if we want to find how many times more intense the level of the concert is over the allowable level, then we need to solve for c and for a.
c/a will tell us how many times the concert level is of the allowable level.
(c-a)/a will tell us how many times the concert level is more than the allowable level.
but first, we need to find the value of c and the value for a.
for the concert, our formula is 105 = 10 * log(c/k).
divide both sides of this equation by 10 to get 10.5 = log(c/k).
this is true if and only if 10^10.5 = c/k.
solve for c to get c = 10^10.5 * k.
for the allowable level, our formula is 100 = 10 * log(a/k).
divide both sides by 10 to get 10 = log(a/k).
this is true if and only if 10^10 = a/k.
solve for a to get a = 10^10 * k.
we now have c = 10^10.5 * k, which is the intensity level of the concert, and we now have a = 10^10 * k, which is the allowable intensity level.
to find out how many times the concert level is of the allowable level, we take c and divide it by a to get c/a = (10^10.5 * k) / (10^10 * k).
the k in the numerator and denominator cancel out and we are level with c/a = 10^10.5 / 10^10.
we solve for c/a to get c/a = 10^.5 which gets us c/a = 3.1623
this tells us the concert intensity level is 3.1623 times the allowable intensity level.
to find out how many times the concert intensity level is more than the allowable intensity level, we subtract 1 from this.
this tells us the concert level is 2.1623 times more than the allowable intensity level.
we can also use the formula of (c-a)/a which becomes (10^10.5 * k - 10^10 * k) / 10^10 * k).
factor out the k in the numerator to get ((10^10.5 - 10^10)) * k) / (10^10 * k).
the k in the numerator and the denominator cancel out and you are left with (c-a)/a = (10^10.5 - 10^10) / 10^10.
use your calculator to get (c-a)/a = 2.1623.
the intensity level of the concert is 3.1623 times the allowable intensity level and 2.1623 times more than the allowable intensity level.
i'm pretty sure this is correct.
give it a shot and see how it goes.
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