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Nomework: Vveek 0 Hiomework (17.0-1 (.0) Score: 0.75 of pt10 of 29 (29 complete)17.6.39-Setup & Solve Find the average temperature on that part of the plane Sx ...

Question

Nomework: Vveek 0 Hiomework (17.0-1 (.0) Score: 0.75 of pt10 of 29 (29 complete)17.6.39-Setup & Solve Find the average temperature on that part of the plane Sx + 2y + 2 = 6 over the square |xl $ 1, ly/ <1, where the temperature is given by T(xyz)=e

Nomework: Vveek 0 Hiomework (17.0-1 (.0) Score: 0.75 of pt 10 of 29 (29 complete) 17.6.39-Setup & Solve Find the average temperature on that part of the plane Sx + 2y + 2 = 6 over the square |xl $ 1, ly/ <1, where the temperature is given by T(xyz)=e



Answers

Find the average temperature on that part of the plane $2 x+2 y+z=4$ over the square $0 \leq x \leq 1,0 \leq y \leq 1,$ where the temperature is given by $T(x, y, z)=e^{2 x+y+z-3}$

Alright, problem 39. I'm gonna find the average temperature on the part of the plane of three x plus for why was e call six where temperatures given by T equals e to the negative Z. It's the first will rearrange function of our plane. Give us six minus three X minus four. Why? Then we differentiate this function with respect to excellence like toe y. So we get Z X equals negative three you Why goes negative four? No, with the fine DS which is you go to squared of Z X squared plus ey squared plus one square and that equals square to 26 d A. Now we can set up our double integral. It's going to be cool double integral of e to the negative six plus three x plus four Why great a 26 d a. Evaluating you can express this integral in terms of x and Y and then evaluating this integral we get or Route 26. Now we want to find the average value so we take right 26/12 and divide that by the value that we got for and that gets us which is the average value given by

So we want to find the average value of a temperature function on a cone and the cone were given a Z squared equals X squared plus y squared where we know that Z is between zero and two. So for a temperature function of Z is equal to 100 minus 25 z. We know that the sphere in parametric form for a conical surfaces are equals our co sign Fada, our sign theta are, um and we know that our is between zero and two and theta is between zero and two pi. So we know that are part conservative arm is going to give us co sign theater sine theta one And then our parson narrative data is going to give us negative r sine theta negative r sine theta, then zero. And, um, we can expect this end of giving us, um when we crossed them. So our cross r theta we'll get a negative are because I'm theta negative. Our sign See you down are and then we take the magnitude of that We'll end up getting is just discriminative too are so now, um t equal to 100 minus 25 z we can write Z as are so have this double integral after i d s the surface in her room. I mean, what wind up getting as a result is 100 minus 25 r d s and this is gonna be from zero to to this will be from 0 to 2 pi and then instead of DS, it'll be, um, times the square root of two are d r d theta. So when we solve this further, obviously the Jew pie will just be off by itself, and then we'll multiply that by the rest of the in the room, which is going to be 1600 route to over six. So we end up getting is just a three. We get 1600 Route two pi over three. Um, also, when we do it with the cross product, So let me do it with this right here. What we're gonna end up getting is four route to pipe. So then, in order to get the average we take, um, the F average going to be the double in a girl service in the room FDs over the double integral DS. So what? That's gonna give us is 1600. Route two Pi over three, divided by four. Route to Hi. We see that the route to is getting cancel and the four will cancel giving us 400. The pie is gonna cancel. So our final answer is going to be 400 over three as our final answer.

So we want to find the average value of a temperature function on a cone and the cone were given a Z squared equals X squared plus y squared where we know that Z is between zero and two. So for a temperature function of Z is equal to 100 minus 25 z. We know that the sphere in parametric form for a conical surfaces are equals our co sign Fada, our sign theta are, um and we know that our is between zero and two and theta is between zero and two pi. So we know that are part conservative arm is going to give us co sign theater sine theta one And then our parson narrative data is going to give us negative r sine theta negative r sine theta, then zero. And, um, we can expect this end of giving us, um when we crossed them. So our cross r theta we'll get a negative are because I'm theta negative. Our sign See you down are and then we take the magnitude of that We'll end up getting is just discriminative too are so now, um t equal to 100 minus 25 z we can write Z as are so have this double integral after i d s the surface in her room. I mean, what wind up getting as a result is 100 minus 25 r d s and this is gonna be from zero to to this will be from 0 to 2 pi and then instead of DS, it'll be, um, times the square root of two are d r d theta. So when we solve this further, obviously the Jew pie will just be off by itself, and then we'll multiply that by the rest of the in the room, which is going to be 1600 route to over six. So we end up getting is just a three. We get 1600 Route two pi over three. Um, also, when we do it with the cross product, So let me do it with this right here. What we're gonna end up getting is four route to pipe. So then, in order to get the average we take, um, the F average going to be the double in a girl service in the room FDs over the double integral DS. So what? That's gonna give us is 1600. Route two Pi over three, divided by four. Route to Hi. We see that the route to is getting cancel and the four will cancel giving us 400. The pie is gonna cancel. So our final answer is going to be 400 over three as our final answer.

Alright problem 42 on the average value of the temperature function T equals 100 minus 25 Z on the Conti squared equals X squared plus y squared. All right, so the do you know where to find the average temperature? We have to calculate the service integral. That one's going to be set up this f times e x squared plus ey squared plus one. So first we need to find ZX. See why those air just partial Ah, partials with respect to Z and why of our function. So we find that z X can be X over a Z and see why to go toe y over Z plugging those into here. We get square to to Okay, now on we set up are integral looking or function for temperature. It's right there. Then we're integrating zero to pi. You're too 100 minus 25 are R T r t data and evaluating this integral we get that simplifies to 800 Route two pi three. So that's what now that we have this value of to divide this by the service area for a cone, which is pi r l and L is gonna be R squared plus eight squared square root That's pie times two times Square to two squared plus two squared that it goes four to Hi. Taking this. Gotta buy that. Get 200 over three as the average value.


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