if-the-inner-and-outer-walls-of-a-container-are-at-different-temperatures-the-rate-of-change-of-te-3

Question

If the inner and outer walls of a container are at different temperatures, the rate of change of temperature with respect to the distance from one wall is a function of the distance from the wall. Symbolically, this is stated as $d T / d x=f(x),$ where $T$ is the temperature. If
$x$ is measured from the outer wall, at $20^{\circ} \mathrm{C},$ and $f(x)=72 x^{2},$ find the temperature at the inner wall if the container walls are $0.5 \mathrm{cm}$ thick.

Stanley Enemuo · Accepted Answer

The temperature of the inner wall can be found by integrating F f X. The eggs, so see put in the integral of F F F so 20 is equal to zero plus C. Sub one. So the expression in crazy on one He is equal to 24 execute Now stuff to tune in general .5 we need to act.

Angela Guo · Answer

were given temperature and Celsius as a function of X and y you were given X and why as functions of time t So we&#x27;re given, um, the partial drogas of temperature with respect to x and y at particular ex wife points. So we&#x27;re asked to find the rate of which temperature changes when time is equal to a wooden second. So let&#x27;s start by finding on the derivative of temperature with respect to time. So we want the t, d t. And Shane will tells us that this is going to be equal to partial drood on the tea with respect to x times d x okay, plus partial the root of the time with respect to Why times d y e t So we know what the party Liotta time with respect to X with respect to why is at ease at this particular point. So we need to find our dxc Tandy White. Let&#x27;s do that here. The x e t. So we&#x27;re given the function of X so the X C T is going to be easy to the power of two T minus two times tip and the Y D T is going to be equal to one over. So now what we want as we want e t d t evaluated at shouldn&#x27;t look was one. So when she is equal to one, when t is equal to one X is equal to eat of the zero, which is one and why is equal to chew plus lawn of one which is zero. So that&#x27;s just too So when t is equal to one, we have X Y is one to which is exactly the point given here. So that means this is gonna be equal to the partial dirty with respect to X evaluated at X y equals one comma too on D x d t evaluated at T equals one plus partial dirt of tea with respect toe Why evaluated at X Y equals one comma too. And times do I d t evaluated a t equals one. So we have these valleys. This is given us as three and this is also given to us. It&#x27;s negative one. So now we just need to find r d x, c T and G u I d t violated At T equals one. And we can do that from the dirt. Is appear. So we plug in t equals one that will give us the X e t evaluated A T equals one is going to be, too. Mmm zero. So that&#x27;s just too Andi Y d t evaluated A T equals one is going to be one over one, and that&#x27;s just one. So which means this is too. And this is one. So now we have This is three times two plus negative one times one. So that is going to be equal to five. I&#x27;ll be five Celsius, her second.

Vikash Ranjan · Answer

doting over the is given asked way, don&#x27;t you? I wanted to look into your power off D minus two in tow to last day when I want to go and go When? A day into one. This is even as three into toe You the power to D minus two. My last one into Wanna work t it is. Hey, don&#x27;t be over. Lofty hat. He was one. Therefore become six about two minus two minus one tinto. It&#x27;s given half $68 0 minus one, which is even as find dignity, sending it to us again. Do that survey given.

Mayukh Banik · Answer

the area over here is eight times for in meter square. The land is 0.2 meter. The hotter temperature is 15 C on the colder temperature, TC is 5 C. We have que as okay a does the tea which is t heart minus t gold divided by the then. And from here you will find that that you will go out to be 1.5. What? Paar meter. They re centigrade. So your college option is D.

Surjit Tewari · Answer

question number 52. The thermal conductivity I would is here the blue Quito zero, born 13. I got four meter kelvin that the American activity off work is 0.46 What are we that Kelvin? If car keys is inside, then let that a picture at the junction B B. We know that the rate off low from the cork and would is equal so we can died. Okay, see into area off one inside temperature 20 minus that junk. Centam bridge 30 upon the thickness off park is equal to thermal conductivity or word into area in Duda off certain nature, my name is zero upon be now substituting the values 0.0 46 indoor when people minus eight is it Cuando 0.13 be But we will solve this. We will get these according to 5.23 degree Celsius. For the be part of the question word is inside. Then we can write cave what into a 20 minus d by the is equal Jake arc into a B minus zero Upon the substituting the values 0.13 don&#x27;t be dynasty is equal to 0.0 hole. Six day solving this, we will get Daisy Caldo 14.77 Big recent. Yes, in the parts. See off the question. It is asked that Does it matter that cork is inside or outside? The answer is it doesn&#x27;t matter because both are good insulators and they will reduce the flow off heat irrespective off their position, no matter.

Brittany Knowlton · Answer

Celsius. Tempter in a region in space is given by t equals two X squared minus X Y Z particles moving in this region. Its position at Time T is given by these three equations. And so, in part, they were asked to find How fast is the tempter experienced by the particle changing and degrees Celsius per meter on the particle is at this point. So you are one unified how fast the temperature is changing. So I know what I&#x27;m wanting is basically my directional derivative which we found in previous problems. And again, that&#x27;s just gonna be the great Nancy times my direction back today. So where are given quite a bit of information to start off with? I know that t of X y Z is just equal to two x squared minus X y z. I also know that my visit, my position at Time T is X equals Tucci squared. Why equals three T and Z equals negative t squared. And I&#x27;m doing all of this problem at the point a six minus four. So I know I need to find these two things. Someone start with my ingredient and I&#x27;m doing that at that 0.86 negative. Four. So to find my great aunt, I&#x27;m taking the partial derivative with respect to each variables to starting with X. I get for X minus getting my wines there now. Constance. Here, sis. Just why&#x27;s he here? Taking the derivative with respect to why this first term goes to zero in here, I just get minus ecstasy and then doing it with respect. Izzie, I get minus X y okay from buying this at 86 Made it for And so when I plug in eight. Frak. Six For why? And negative four for Z into this equation. I see that I get 56 I plus 32 j minus 48. Okay, so I found my 1st 1 now, and he would find my direction vector. So again, if you recall my direction, vector is just equal to the busty specter over the length of it. So first, to find this, I need to start with my position. So I know that my position is these three. So change us into my vector. Our tea. Basically, these are just gonna come the coefficients of I J and K respectably. So I get to t squared I plus three t j minus t squared. Okay, so to find my V, I&#x27;m just taking the derivative of this with respect to t. So I take the derivative I get for T I plus three j minus two tea. Okay, so I want this and for the position. I want this to be at the point a six negative four. So, basically, I need this coefficient to equal eight. I need this coefficient equal six. And this one&#x27;s the equal negative for So I see if I were to princess sent this sequel. Six obviously, t is equal to two. But what people said to here to get my eight and I do get my negative for here. So then just to make a note of that, that tea cools to particle uh, 86 negative for So I&#x27;m gonna find what this is at my position to assess where we are finding this problem. So is good. Hey, I plus three j minus working. So I now have my the vector at the position that I want for this problems. Now I confined my you So it&#x27;s just going to equal this new B a to. And so if you find that you should get a over the square root of 89 I plus three over square root of 89 J minus four over square root of 89. Okay, so you can see these are just the same coefficients from here. So length of the vector was sweet. Group of 89 Oh, it&#x27;s so have my you I have my Grady int. So now I can find my directional vector So again, this Do you have tea at the point it&#x27;s one equals to this Grady in a t dot My you. So that&#x27;s gonna equal to just pulling out that one over the square root of 89 to the front. It&#x27;s going to be all these coefficients of the same I, J and K multiply together 778 10 56 When I take the jays, I get 32 times three and then the case I get a, um, negative 48 times negative for And so when you multiply that all out, you should get seven over 36 over a square root of 89 degree Celsius from eater as your answer. Okay, so now for part B. We it says How fast is the temperature experienced by the particle changing in degree Celsius per second. So now we are looking at our main equation again this see So to find the change, it&#x27;s going to be d big t over little T. So if I take that derivative after first, take the partial derivative with respect to X and then times the ex duty to get this d t over DT plus the partial derivative with respect to why times, d y d t And so looking at this, I do see some commonalities. I know this comes from the Grady int, which was our coefficients from here. I know that this DX dt came from my be over here, so I know that this is just going to be the same thing as the grating and t dot might be Vector which to get this in terms of you is just the same thing as Grady Antsy. I got my you vector times the length of B. What were you found? This one. So At T equals two. We already found that we know that this is just going to be people to 7 36 over a scribbler of 89. And I know that&#x27;s just gonna be this Gregory of 89. So this is just equal to 7 36 degrees Celsius crew. Second and again, this is because the T T is equal to do you thi At T equals two victor.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

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