5. Sound intensity is most often measured in decibels (dB)...

5. Sound intensity is most often measured in decibels (dB) according to the formula $D(x) = 10 \log_{10}(10^{12}x)$, where $x$ is the intensity of sound in watts/cm². (a) Ordinary conversation occurs at $10^{-6}$watts/cm². What is the intensity in dB? (b) Noise levels above 70dB will harm the hearing after repeated exposure. What is the intensity of sound in watts/cm² that will harm the hearing in this manner? (c) For any $x > 0$, calculate the difference $D(2x) - D(x)$.

Transcript

00:01 In this question, a formula for relative loudness of the sound that is d is given as 10, log, i divided by i0, where i is the intensity of the sound in decibel and i0 is the standard threshold of audibility.

00:18 And now, first of all, actually there are three parts in this question, but first of all, we will solve this equation for i intensity.

00:27 Okay and now we will divide both side of the equation by 10 so it will be 10 divided d divided by 10 equals to log and i divided by i not okay and we know the base of log is 10 and we know a property of logarithm that is if log base b x equals to y then the value of x is b raised to the power y okay so now we will use this so base is 10 here and instead of x there is i divided by i not okay so from here value of i divided by i not it will be 10 raised to the power d divided by 10 and now we will multiply both sides of the equation by i not so i will be i not multiplied by 10 raised to the power d by 10 okay so this is the formula for intensity okay and now we will use this formula of intensity to find out the parts a b and c so part a we have to find out intensity okay at 30 decibel sound so d is given 30 so we will put d equals to 30 in our formula okay so it will be i equals to i not okay and 10 raised to the power d by 30 that is 30 by 10 or we can say 3 so this will be the intensity at intensity at 30 decibel sound okay we will mark it i 30 okay normally we will mark it i 30 and now part b of the question that is how many times greater the intensity of 80 decibel sound is than of 30 decibel town okay it means we have to find out i 80 divided by i 30 okay intensity at 80 decibel sound and intensity at 30 decibel sound so i 80 it will be i not okay from here it will be i not 10 raised to the power 80 divided by 10 that is 8 divided by i 30 we have already find out i not 10 raised to the power 30 by 10 that is 3 and when we solve it will be 10 to the power 5 and it 10 to the 4 5 times greater okay so the answer of part b will be 10 to the 5 times greater than okay and now part c that is that is prolonged noise above 150 decibel cause permanent deafness so how does the intensity of 150 decibel comparable with the intensity of 80 decibel? so we have to intensity of 150 comparable of intensity of 80 decibel.

03:08 So we will find out the value from the formula.

03:11 I equals to i0 10 raised to the power d by 10...

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