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Three sheets of plastic have unknown indices of refraction. Sheet 1 is placed on top of sheet 2, and a laser beam is directed onto the sheets from above so that it strikes the interface at an angle of $26.5^{\circ}$ with the normal. The refracted beam in sheet 2 makes an angle of $31.7^{\circ}$ with the normal. The experiment is repeated with sheet 3 on top of sheet $2,$ and with the same angle of incidence, the refracted beam makes an angle of $36.7^{\circ}$ with the normal. If the experiment is repeated angain with sheet 1 on top of sheet 3 , what is the expected angle of refraction in sheet 3$?$ Assume the same angle of incidence.

$\text { The angle of refraction in sheet } 3 \text { is } 23.1^{\circ}$

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Cornell University

Numerade Educator

University of Washington

Hope College

so far our question. We are asked to consider a breach sheets of plastic with unknown index, every fractions. But it tells us the angle of incident as well as the angle of refraction for the boundaries. For the 1st 2nd and third sheet here were asked to find the expected angle every fraction for she three using all this information. So we're gonna break this into three parts. We're gonna find the relationship for the index of refraction for in 12 into as well as in, um, in three toe into and then using this relationship, we can go ahead and cancel out these index of refraction. Since we don't know them to find our final angle, let's go ahead and work through that. So for the first part, in one times a sign of the incident angle 26.5 is equal in two times the sign of the outgoing angle 31.7 or in other words, in one is equal to into times the ratio of those two signs. So we're just gonna use s here to indicate signs. So any time I write s, that means sign so we don't drive, sign every time. So this is sign of 26.5 degrees. Well, go then. Put parentheses around this divided by the sine of 31.7 degrees. Okay. And then using that exact same logic for the second scenario, we have ah, in three this time is going to be equal to into. And then again, we're giving the angles. This time it's the sign of 36.7 degrees, divided by the sign of 26.5 degrees. Okay, so now that we have relationships between in one and three and in two, we can go to our third part here, where in one times the sign of data one which in this case is 26.5 degrees, is equal to and three timesthe sign of the unknown angle data that we're trying to find. Okay, well using now the relationship between in one and into and then three and into we can put everything in terms of index of refraction of the second piece of plastic, cancel it and make this solvable. So now, replacing in one with into times the relationship of sign of 31.7 degrees divided by the sine of 26.5 degrees. Of course, this is all still multiplied by the sign of 26.5 degrees. So we can see that those were going to cancel. And then now replacing in three within two times the ratio of the sign of 31.7 Excuse me, 36.7 degrees, divided by the sign of 26.5 degrees, all still multiplied by the sine of the angle. Fada, we're trying to find. So the in twos cancel here, cancel those out. This sign of 26.5 canceled this sign of 26.5 and then solving for data, we find that data is equal to the inverse signs will write that as s to the minus one of the sign of 31.7 degrees multiplied by the sign of 26.5 degrees. And this is all divided by the sine of 36.7 degrees. So taking the inverse sine of all of that, we find that data is equal to 23 0.1 degrees weaken box that in as the solution to our question