In a pig-calling contest, a caller produces a sound with an intensity level of 110 dB.
How many such callers would be required to reach the pain level of 120 dB?
To determine the number of callers required to reach a certain intensity level, we can use the fact that the intensity level doubles for every increase of 3 dB.
First, we need to calculate the intensity ratio between the pain level of 120 dB and the initial level of 110 dB:
Intensity ratio = 10^((120 dB - 110 dB) / 10)
= 10^(10 / 10)
= 10^1
= 10
This means that the intensity level at the pain level is 10 times greater than the initial level. Therefore, we need to find how many times the initial intensity level needs to be doubled to reach this new level:
10 = 2^n
Solving for n, the number of times the intensity level needs to be doubled:
n = log2(10)
≈ 3.3219
Since n represents the number of callers required, we round it up to the nearest whole number to get the final answer:
Number of callers required = ceil(n)
= ceil(3.3219)
= 4
Therefore, we would need 4 such callers to reach the pain level of 120 dB.
To determine the number of callers required to reach the pain level of 120 dB, you need to understand the concept of combining sound intensities.
When combining sound intensities, you can use the formula:
I_total = 10 * log10(N)
Where:
- I_total is the total intensity level in decibels (dB).
- N is the number of sound sources.
First, let's convert the 120 dB pain level into intensity units using the inverse of the formula above:
I_total = 10^(I_total / 10)
= 10^(120 / 10)
= 10^12
Now, let's substitute this value into the equation and solve for N:
10 * log10(N) = 12
Divide both sides by 10:
log10(N) = 1.2
Taking the inverse logarithm of both sides:
N = 10^1.2
≈ 15.85
Since you can't have a fraction of a caller, you would need at least 16 callers to reach the pain level of 120 dB.
Going to 2 callers will add 3 db to the sound level, which is on a logarithmic scale. Doubling to 4 will add 3 db more.
Eight will add 3 db, making a total of 119 dB. You need ten callers to add 10 dB to the sound level of one caller.
That should have been obvious from the definition of dB.