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Newton's Law for conduction of heat says thatKT:for some constant * > 0.Suppose that when the external temperature (T:) Is degrees higher than the internal ...

Question

Newton's Law for conduction of heat says thatKT:for some constant * > 0.Suppose that when the external temperature (T:) Is degrees higher than the internal temperature the Internal temperature is changing at 0.6 degrees per second. Find the rate that the internal temperature changing when the external temperature Is 13 degrees higher than the Internal temperature;Flnd the rate that the internal temperature changing when the external temperature Is 13 degrees lower thanthe Internal temper

Newton's Law for conduction of heat says that KT: for some constant * > 0. Suppose that when the external temperature (T:) Is degrees higher than the internal temperature the Internal temperature is changing at 0.6 degrees per second. Find the rate that the internal temperature changing when the external temperature Is 13 degrees higher than the Internal temperature; Flnd the rate that the internal temperature changing when the external temperature Is 13 degrees lower thanthe Internal temperature; Find rhe rate that the Internal temperature changing when the Internal and external temperatures ane equal,



Answers

Newton's law of cooling describes how a physical object comes into thermal equilibrium with its environment. This law can be expressed as $H L=\mathrm{K}\left(T_{\mathrm{o}}-T_{\mathrm{a}}\right)$ where $H L$ is the rate of heat loss, $K$ is the thermal conductance constant (how easily an object loses heat), $T_{\mathrm{o}}$ is the temperature of the object, and $T_{\mathrm{a}}$ is the ambient temperature. Apply this expression to the metabolic rate/ environmental temperature curve for endotherms (see Figure 40.17 ). What would be the equivalent of $\mathrm{HL}$ ? In Newton's law of cooling, $\mathrm{K}$ is a constant reflecting the properties of the object. What would K represent for an endotherm? What would 1/K represent? Using a version of Newton's law that replaces $T_{\mathrm{o}}$ with $T_{\mathrm{b}}$ (body temperature), explain why the curve projects to zero at an ambient temperature that equals body temperature.

So we give it a different question of Did see or C d. C. He called. So Kate off m minus c I write. Rewrite this as did see over Did see you call to minus K. I pull out of miners and I have C minus AM This is just the same thing. Now I separate these and I have did see Yeah c minus aim a call to minus Kate multiply by dizzy now Lester the integral of both sides. You have that the integral of this would just be the natural look of absolute front of C minus aim he called. So when integrate these cases constant. So you have minus Kate, see? And then let our plus one. The next thing we do know is that would sink the exponential of both sides. I have eaten and I have e so that gives me that's this would cancel out on half. That's C minus m. You called tau eat the minus, Kate. See, multiply by e to the C one, which is the call to If I said this either to see one is a constant, so I can name that sits suit. So I've seen to eat the minus key. See, So that implies that absolute vial of C minus aim you called to seed soup e minus can see which implies that C minus AM 1/4 plus a miner's off sit su e minus kid. See, this implies that See, it called. So am I lost C E minus key. See, we're C equals plus minors off. See Sue. So let's named execution style. We're gonna use a litter on now. The B pots we're given That's the constant temperature. Am equal to 70 degrees were given C at 08 quarts 100 degrees were given C at six. You called to 80 degrees and we want to find see our twin. See, that's what we're looking for now plug in this into the equation. Star plugging into star The questions start with God's. So we have that seem at zero that's on dread. Equal to 70 are any 17 plus c e t the because as equals zero does it zero. Andi also has eight. We have eight c equal to I will on Emma's to 17 last see each day not equal to six. That's my No. Six cakes. Now the first part implies that's because this is dessert. Has warned that implies are seen. It quotes with 30. Now, this would imply that a C minus $70. 10 it wants of that c e to the minus escape. This implies that eatery minus six K you called. So 10 of the 13 which is one over three. No, I want to solve for case. So what I do is that I think natural look of both sides and then that implies that this would cancel out. I have doubts. Minus six, Kate according to natural law off one over three, which implies dubs que quarts of minus six. So remind us one of the six that's minus one of the six off. Let me write these the better way. So I divide with sites of these by Negative said So I have cake. What's negative? One over six. Not sure a load off one over three. So this implies that's back to star in start. I'll see equal to m. My m is 70 on then, plus c eat. Okay, go by sea to be 30 also. So let me plug that thing. Not steady Eataly minus king So they'll be minus off minus one of the six. Not sure Local one of the three. So this becomes plus rights. So that is the court. So, um, under does it see I'm missing? It seems more supply by sea because that's minus key scenes. There were more supply base by sea. I haven't seen here. It's about 70 plus that c That's let me just write into the better we eat and see over six natural off one over three. Now I can pour this city this six upwards using the laws of modern things. So I have that seeing equal to 70 lost at sea when the let's go about darts would have e to the natural of off one of the three to the power of one of the six nuts eat in natural law off one of the three to the policy over six. So this would cancel out. Now I'm looking for C Watts, sweetie. See, I said, I'll be 70. Lost Mort. Supplied by one over three off. 20 over six. When blood is in, you have 73.772 gates

Okay, So for this problem were given the equation f of tea equals 18. I'm sorry. We're given equals t o plus C e to the negative, Katie. And so we're given that t is nine. Were given that teen not is. 18 were given that CS five. And that RK is 0.6. So negative 0.6 and were asked to find out What's the, um The temperature of the object is okay, So all we have to do is salt for this. So by utilizing a calculator, we find out that it is 18.23 degrees.

Okay, So the temperature of the surroundings is given as let's call this T sub as the temperature of the surroundings eyes going to be 60 degrees Fahrenheit and the thean initial temperature difference. So let's call that Do not be sub zero is equal to well in normal body temperature of 98.6 minus 60. So this D sub d subzero is gonna be equal to 38.6 degrees Fahrenheit. Okay, so now, by Newton's law of cooling the temperature after t minutes is going to be modeled by dysfunction here. Um, so capital T here t of tea is gonna be equal to Well, the T's best was 60 and then plus the T subzero, also plus 38.6 on den times e raised to the negative K. T. Um, Okay, well, we're given that K here is equal to 0.1 947 So therefore, um, we have that our temperature is given by TFT is going to be equal to 60 plus 38.6 times e raised to the negative, uh, to the negative 0.1 947 t. All right, so there is our my own for hard a not of part B. We use the function we found in part a here, um, with while TFT now being equal to 72 we solve this equation for tea. So we just go ahead and we set this equation equal to 72 then south for tea. So we have 72 is equal to 60 plus 38 0.6 times e raised to the negative. Zero 0.194 70. Okay, well, t is up in the exponents here, right? We have e raised to a policy to undo this. To get the tea, we're gonna take the natural log off both sides. So what do we get? Well, we get the natural log of off the right side again. Brings the next lot of e to the power is just the power. So what we end up with here is, um Well, we get 38 point six times ive raised to the negative 0.1947 t is groups is going to be equal. Um too. Well, 70 to minus 60. Okay, so we get So then take the next one of both sides we devote divide through by the 38.6. Just get e to the negative. 0.194 70 is gonna be equal to about 0.31088 ish. Okay. And then taking the natural log off both sides. Well, the natural log of veto power just gives the power backs. We just get negative. 0.194 70. Is that equal to the natural log off the right side. That's natural. Log off. 0.31088 08 to 9. Okay, that again? Subtracting here, Dividing through by 38.6. And it's all for T. We just divide through by zero. My negative. 0.1947 So that gives us that T is going to be, um, equal to about 6.0 zero, I think 747 we get So we can say here that t is approximately equal to six. Right? So therefore, a time of death, the temperature of the all the that, um, the time here right is gonna be six hours, right? Because we're approximately approximately equal to six. So we get that tea is well, approximately equal to six, um, hours

Okay, So the question was asking us Holland, where we take the house temperature to reach the reason port so we could determine the timing. Okay, so we know that he lost Where can we go? That negative k a has DT over dx isa temperature acts is the, uh the thickness. Okay. And in case the thermal conductive, he is the surface area. We know that. Did he come express as changing temperature, which is t minus T zero t is the temperature of the rich tea's roasty initial temperature, which is the outside temperature, which is on Sorry, I should pull out 1915. Okay, so that teaser was negative. 15. The results is okay, Arthur Thermo resistance. So now Okay, T t o d Actually, see, it was a negative. T my teacher of our people t minus t zero Varzi bouquet times tt over the axe. Okay. And we also know that the heat loss rate can express the heat over per unit. Time was a dick. Your duty which can be also it was never t Manistee zero of our way. Do some arrangement here will get take you over the future. Some people take you over. DT Times Capital DT over DT So it's see chastity over TT See, here's a he capacity. Okay, so who have seen times tt over the lower case T which is equal to negative team honesty. Zero r sees the vehicle as it was given a six point by May God you her Calvin, which is six point by tens Tension Power six to help Thermo resistant ar is 6.67 Milly Calvin for Watts, which is 6.67 times 10 to her apartment 83 Calvin for what's if we do some arrangement would have DT on left side And, uh, the lower case. He wishes the tie on rice. So it's DT over t minus t zero is ableto connective wall over cr DT. So now let's put it into your own. We have t to ah is a She's what you want, OK? And this will have zero to t. Okay. Hey, things grow here. Who have national log? T minus t zero from t two t one, which is you go to Yeah, minus t oversee our arrange is wrong. Zero to t. Okay, so now every you plugging Uh So the tea here? No, it's never showed us. So it's natural. Rt minus 15. Um, ser minus negative. 15 degree Celsius. Remember t zero sq as 15 degrees houses. And he should be 20 degrees Celsius and he wants with zero degrees Celsius. Why is that? Because, remember, the question was asking us. Hello? Will you take the house members temperature to reach the freezing points? So it's from 23 results. It's zero degrees Celsius. Okay. And on the right side, just simply Ah, negative t oversee our because if you're probably zero, here is zero. So just 90 Okay. And the question wants to determine times wanting to move the T on laps lower. Tasty. Okay, so let's do it. So we have negative oversee. Our is equal to natural log T plus 15 degrees Celsius. This minus negative. 15 years plus 15 degrees houses. Okay. In front. 20 degrees houses to zero degrees houses. Okay. They will have minus t. Um, oversee. Our is equal to natural log zero plus obtain degree sashes over 20 degree Celsius plus 13 degrees houses. Okay. Why is that? I want to rest. So if it is based on the sea question. So I say natural log X, my nose natural Agwai. And this will be able to natural log axed. Remember Why? Okay, this is from nausea, so I love you. Carried over. Okay, so I use is formula here to get natural. Log zero plus 15/20 plus 50. Okay, It's a level that's keep doing our incursions. Here they will have. Tea is equal to inactive. Natural log. They're thing degree Celsius over 35 degrees houses. Okay. Over. Almost are no over time. C r. Okay, that's plugging and see value and are very well have teas. Eagle two, Nigel log 15 over 35 since degrees, houses and degree Celsius. Cancel out. Okay, times see was given. As you see, 6.5 times 10 to us. Power six juco Kelvin. Okay, six point fight ends 26 Tuber Calvin times 6.67 times. And to elect three, uh, Calvin her watts. Okay. And this will give us a times about 300 I'm sorry. 36,000 and 734 0.6 seconds. Every converting our which is, uh, 36,734 0.6 times One over. Over 30. 600. Okay, because what are their 36 100 seconds? Okay, so therefore will have, um is about 10.2 hours. Okay, so you take about 10.2 hours to take the house temperature to reach the freezing point. Okay? And this is my answer for the question. Thank you.


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