(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity

Video transcript

All right, everyone. So for this problem, we are working with sound intensity level and it's a two part problem. And the first part is actually kind of a, a proof um working with the equation for sound intensity level. So it's asking why it's possible to determine the multiplication factor for sound intensity that increases sound intensity level by 22 decibels without knowing the initial sound intensity, let's prove why that's possible. And then the second part is finding the numerical value of that factor. OK. So the first thing we need to do for this problem is recall the equation for sound intensity level. So delta beta equals 10 decibels times the log of I two over I one knowing that the uh the multiplication factor or the sound intensity that increases the sound intensity. So think about this, the multiplication factor for this sound intensity is going to be I one times X sum multiplication factor where this initial sound intensity is just I one. So we can rewrite I two as X times I one, we plug that back into this equation. Our change in sounds sound intensity level. It's 10 decibels times the log of X I one over I one I won that initial initial sound intensity cancels. And so you're left with this equation where as long as you know the difference in sound intensity level, you can find that multiplication factor. So the reason for the first problem, the answer to the first part of this problem, why is it possible is because the ratio I two over I one does not require the value of I one. So when we're looking at these uh potential answers here, we can uh eliminate some of these answers right off the bat, right? It's all about the ratio, not the difference. We do know the difference in uh sound intensity level. It's given to us is 22 decibels, it is an increase. So it's positive. So 22 decibels equals 10 decibels times the log of X. And we're just gonna solve this equation. Um We're gonna uh divide both sides by 10 decibels. So the 2.2 equals the log of X, we need to remember some log math here that can be rewritten as X equals 10 to the 2.2. And that's in our plug that into our calculator and that is 100 and 58 decibels. So now we look back at our answers, the ones that are still left and we get answer A, the ratio does not require the value of I one. And the numerical value of the factor is 158. All right, that's it for this video. See you in the next one. Thanks folks.