An 8 hour exposure to a sound level of 95.0 dB may cause hearing damage. What energy in joules falls on a 0.850 cm diameter eardrum so exposed?
To calculate the energy falling on the eardrum, we need to use the formula for sound intensity:
I = P/A
where I is the sound intensity in watts per square meter (W/m²), P is the power in watts (W), and A is the area in square meters (m²).
Given that the sound level is 95.0 dB, we can convert it to sound intensity using the formula:
I (in W/m²) = 10^((L - 12)/10)
where L is the sound level in dB. Plugging in the values, we have:
I = 10^((95 - 12)/10) = 10^(83/10) = 10^8.3 ≈ 1,995,262.31 W/m²
Now, we can calculate the power using the formula:
P = IA
where A is the area of the eardrum. The diameter of the eardrum is given as 0.850 cm, so the radius (r) can be calculated as half of the diameter:
r = 0.850 cm / 2 = 0.425 cm = 0.425/100 m = 0.00425 m
Using the formula to calculate the area of a circle:
A = πr²
where π is approximately 3.14159, plugging in the values, we have:
A = 3.14159 * (0.00425)^2 ≈ 0.000056709 m²
Finally, we can calculate the energy falling on the eardrum:
Energy = P * t
where t is the exposure time. Given that the exposure time is 8 hours, we need to convert it to seconds:
t = 8 hours * 60 minutes/hour * 60 seconds/minute = 8 * 60 * 60 seconds = 28,800 seconds
Plugging in the values, we have:
Energy = 1,995,262.31 W/m² * 0.000056709 m² * 28,800 seconds ≈ 32.62 joules
Therefore, approximately 32.62 joules of energy falls on the 0.850 cm diameter eardrum exposed to an 8-hour sound level of 95.0 dB.