You are trying to overhear a juicy conversation, but from your distance of 15.0 , it sounds like only an average whisper of 35.0 . So you decide to move closer to give the conversation a sound level of 60.0 instead. How close should you come? (in cm)
You need to get closer to the speaker so that the sound power per area is increased by a factor 10^2.5 = 316. According ti the inverse square law, this will require getting closer by a factor sqrt (316) = 17.8. This will put you just abut 1 cm away from the speaker's lips.
To solve this problem, we can use the inverse square law of sound propagation. According to this law, the intensity of sound decreases as the square of the distance from the source of the sound.
First, let's calculate the initial sound intensity (I1) from the given whisper level of 35.0 dB. We convert the decibel scale back to the intensity scale using the formula:
I1 = 10^(dB/10)
I1 = 10^(35.0/10) = 316.22777 units (approx)
Next, let's calculate the final sound intensity (I2) required to have a sound level of 60.0 dB. Again, we convert the decibel scale back to the intensity scale:
I2 = 10^(dB/10)
I2 = 10^(60.0/10) = 1000 units
Now, we need to find the distance (d2) closer to the source that would result in the desired sound level of 60.0 dB. We can rearrange the inverse square law equation to solve for d2:
I2 = I1 * (d1^2 / d2^2)
d2^2 = (d1^2 * I1) / I2
d2 = sqrt((d1^2 * I1) / I2)
Given that the initial distance (d1) is 15.0 meters (1500 cm), we can substitute this value along with the calculated values for I1 and I2:
d2 = sqrt((1500^2 * 316.22777) / 1000) = 536.656 cm (approx)
Therefore, you should move closer to a distance of approximately 536.656 cm to have a sound level of 60.0 dB.