Use the following discussion: The loudness L(x), measured in...

Use the following discussion: The loudness $L(x),$ measured in decibels $(d B),$ of a sound of intensity $x,$ measured in watts per square meter, is defined as $L(x)=10 \log \frac{x}{I_{0}},$ where $I_{0}=10^{-12}$ watt per square meter is the least intense sound that a human ear can detect. Determine the loudness in decibels of each of the following sounds. Diesel truck traveling 40 miles per hour 50 feet away. intensity 10 times that of a passenger car traveling 50 miles per hour 50 feet away whose loudness is 70 decibels.

Use the following discussion: The loudness $L(x),$ measured in decibels $(d B),$ of a sound of intensity $x,$ measured in watts per square meter, is defined as $L(x)=10 \log \frac{x}{I_{0}},$ where $I_{0}=10^{-12}$ watt per square meter is the least intense sound that a human ear can detect. Determine the loudness in decibels of each of the following sounds.
Diesel truck traveling 40 miles per hour 50 feet away. intensity 10 times that of a passenger car traveling 50 miles per hour
50 feet away whose loudness is 70 decibels.

Key Concepts

Decibel Scale

The decibel scale is a logarithmic unit used to measure sound intensity relative to a reference level. It simplifies the representation of the wide range of sound intensities by converting multiplicative differences into additive values, making it easier to compare sounds.

Logarithmic Function

Logarithms are fundamental in converting the ratio of two quantities into a more manageable number. In sound measurement, using a logarithmic function helps express the exponential range of sound intensities as a linear scale, which aligns better with human perception.

Reference Intensity

Reference intensity is a standard value against which other sound intensities are measured. In acoustics, it is typically set at the threshold of human hearing, allowing for a consistent and meaningful calculation of sound levels in decibels.

Transcript

00:02 We're given a loudness function, l of x, where l is the loudness in decibels and x is the intensity of the sound.

00:12 I not is the intensity of the least intense sound that a human can hear.

00:18 Furthermore, we're given that for a car, l of x of c is 70 decibels.

00:29 So that would have x of c is the car intensity.

00:35 And we're given also that we have a truck, we'll call his intensity x sub t, that is 10 times the car.

00:45 So it's 10 times x of z.

00:48 And what we want to find is the loudness of that truck.

00:53 Now, you could reason it out that if we're going to increase the x by a factor of 10, and this is log base 10, log base 10 of something increased by a factor of 10 is increasing that log by one.

01:11 So if the car is 70 decibels, then the truck must be 80, right? because 7 plus 1 is 8, and then when you multiply it by this 10, you get 80.

01:19 So you can reason that out with logic.

01:22 But let's go ahead and do the math for the fun of it.

01:27 So we have that l of the car's intensity is 70.

01:33 So we can start with that and say we know that 70 equals 10 times the log of x over i not.

01:45 This is x subcar, it's not 10, x of i sub zero.

01:53 And so that if we divide both sides by 10, we get 7 equals the log of x sub c over i not.

02:03 If we want to get, if we want to know what x of c is, we're going to have to exponentiate.

02:08 So remember if there's no log, a base written next to the word log, it's implied to be 10.

02:14 It's the common log.

02:16 So if we want to exponentiate this, we start with a base, come around and grab the other side as our exponent, and then come back around and grab the insides.

02:25 So the car's intensity is over i0 equals 10 to the seventh...

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