Question

The loudness L(x) measured in decibels, of a sound of intensity x, measured in watts per square meter, is defined as L(x)=10 log (x/I base 0=10^-12 watt per square meter is the least intense sound that a human ear can detect. Determin the loudness, in decibels, of each following sounds. 1. Diesel truck traveling 40 miles per hour 50 feet awar: intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away whose loudness is 70 decibels

Answer 1

To solve this problem, we need to first find the intensity of the passenger car traveling at 50 miles per hour, which is given to be 70 decibels. We can use the formula L(x) = 10 log(x/I_0) to solve for x:

L(x) = 70 dB

10 log(x/10^-12 W/m^2) = 70

log(x/10^-12 W/m^2) = 7

x/10^-12 W/m^2 = 10^7

x = 10^7 * 10^-12

x = 10^-5 W/m^2

Therefore, the intensity of the passenger car traveling at 50 miles per hour is 10^-5 W/m^2.

Next, we can find the intensity of the diesel truck traveling at 40 miles per hour and 50 feet away, which is 10 times that of the passenger car. We can use the inverse square law of sound to solve for the new intensity:

I1 / I2 = (d2 / d1)^2

where I1 is the new intensity, I2 is the original intensity (10^-5 W/m^2), d1 is the distance from the passenger car to the listener (50 feet), and d2 is the distance from the diesel truck to the listener (also 50 feet).

I1 / 10^-5 = (50 / 50)^2

I1 = 10^-5 * 1^2

I1 = 10^-5 W/m^2

Therefore, the intensity of the diesel truck traveling at 40 miles per hour and 50 feet away is also 10^-5 W/m^2.

Finally, we can use the formula L(x) = 10 log(x/I_0) to find the loudness of the diesel truck in decibels:

L(x) = 10 log(10^-5/10^-12) = 70 + 10 log(10)

L(x) = 70 + 10

L(x) = 80 dB

Therefore, the loudness of the diesel truck traveling at 40 miles per hour and 50 feet away is 80 decibels.

Explanation: