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04 A metal bar at a temperature of 1000 F is placed in a room at a constant [10] temperature of 10* F. If after 15 minutes the temperature of the bar is 50*F Find (...

Question

04 A metal bar at a temperature of 1000 F is placed in a room at a constant [10] temperature of 10* F. If after 15 minutes the temperature of the bar is 50*F Find () the time it will take the bar to reach a temperature of 25" F the temperature of the bar after 10 minutes.

04 A metal bar at a temperature of 1000 F is placed in a room at a constant [10] temperature of 10* F. If after 15 minutes the temperature of the bar is 50*F Find () the time it will take the bar to reach a temperature of 25" F the temperature of the bar after 10 minutes.



Answers

A cold metal bar at $-30^{\circ} \mathrm{C}$ is submerged in a pool maintained at a temperature of $40^{\circ} \mathrm{C}$. Half a minute later, the temperature of the bar is $20^{\circ} \mathrm{C}$. How long will it take for the bar to attain a temperature of $30^{\circ} \mathrm{C} ?$

So we are given Newton's law of cooling and were given the initial information that is underlined in red. And with the only reason we're given that information is so we can find case. So let's work that out over here. 35 -70 is -35. And that's going to equal 28 -70, Which is 42. Actually it's negative 42 E. To the turn K. So we moved the 70 over, get negative 35 and then we have negative 42 E. To the 10-K. So we have 35 all over 42 E. Equals E. To the 10-K. And if you want to reduce that more, that's fine. The natural log of 5/6, you could put 35/42 in there equals 10. Uh huh. So the natural log Five divided by six equals and I want to divide that by 10. And I have a negative .0182, negative 0.182 is K. And that's what we're gonna use from k. From that point on first question Temperature after 30 minutes. Well, That means we are looking for 35 equals. Nothing changes up here because and we still have we can put that already in there -42 E. To the negative .0182 times 30. And that's not what I want is the 35. That's what I'm looking for. The temperature after 30 minutes. So I'm putting 30 and for tea I'm using the K. That we found and it's still 70 -42 equals. So let's do that 70 -42 e. race to the negative .0182. And I can't remember how many minutes 30 minutes times 30. So that temperature is 45.7°.. Or no temperature, Yep. Let's pick a different color and how long for it to reach 45°.. So we're going to put 45. And for that Again, the ambient temperature is still 70. The original temperature is still 28. And we have E. To the that's what R. K. In there,- .0182 T. So we have 45 -70 again is negative. That's 25. Negative 25 Equals -42 E. To the negative .0182 T. 25 -42. or divided by -42 Equals each a negative .0182 T. The natural log of 25/42 Equals negative .0182 T. So the natural lock Of 25 200, Yep. 25 divided by 42 that equals app. And we want to divide by back. Mhm. Yeah And we get 28.5 minutes. Not bad. Yeah.

Uh huh. So last sight t zero equals to 40 degree and we'll have a function white t eco's to 40 plus See time's U to the power off minus Katie. So when t equals to zero, why here equals to minus 30. So why the arrow Eco's to 40 plus c equals two minor 30. That means our see here equals two minus 70. And then after 30 seconds, that means why 30 equals two for T minus 72. The power off minus 30 k equals to 20. So we'll have our k here equals minus 1/30 law into our seven that approximately ecos to the rope on for two. So to attend a temperature of 30 we are so that 40 minus 70 e to the power off minus 700.42 Tom's t equals to 30. So to solve that, we can see t here just equals to loan 1/7. Our minor 0.4 to wish approximately equals two 47 seconds right

Broken in number 26 uh, which any April banks is it with dean? Fines 30 minus 15 over 10 X. Jeez, Nick, we're through 15. Close 1.5. Um, this is the temperature at position. Next ah were F average, which is the one over B minus minus A. So it's 1/10 from zero. Then look for a and to be off every labs, which is 15 plus one point by X. Yet so the president answer for that would be up to the integration is went through fire over then, which is 24.4, which would be F or X store. This is equally true unless 1.5 x is equal to 22.5, just the if average so X is equal to 7.5 over 1.5, which is equal five liters since the park is 10 m average. Richard is thus obtained in the middle off the bar, which is 5 m, so it submitted of there for so by the mean many of hearing for integration that has to exist a point on the floor where the temperature is the same as is a

Do the law of cooling the temperature of an object at Time T is governed by this differential equation. So we have Who's VT? Um, d t is equal to negative. Okay, times well, capital t minus t sub em Minus t's to them. Okay, so what? Um well, so t's Obama is the given temperature of the room. So here were given that Tisa bam is 75 degrees. Okay? And so we want to substitute 75 4 t sub em in. Well, um, this equation here to get Kiki U T is equal to well, negative k times t t minus 75. Okay, so now we just rewrite this equation by separating our variables. So what we get is, well, one over t minus 75. Um, e t we separate our casualties and small teas and, um is equal to well, negative k g t. Okay. And then what you separate are variables. We can then integrate the equation, so we just integrate both sides. That inner girl is equal to that integral. And what we get is well, this left side hand side becomes just the natural logs of the absolute value of T minus 75. Okay. And that is equal to negative, Katie, plus your constancy cash. So now it is given that the initial temperature off the object is 615 degrees. So, um, we get that while 6 15 is equal, six 15 is equal to 75 plus seats. Call it C one. So therefore we'll see. One is then just equal to 6 15 miles. 25 or 5 40 is then equal to see one. Okay, Now we can subject to the value of C one in, um, t of teas are function capital. T of tea is equal to 75 plus. See someone's e to the negative. K t. What we get is that well, TFT, um, is equal to still, this is equal to 75. Ah, plus 5 40 e to the negative. Katie. All right, um, so then let's let the time for the temperature of the object, um, be 100 and 35 F. Um, let that be equal to x hours time for the temperature of the object to be 100 and 35 F, which is equal to x hours. Then we have we have 135 is equal to 75 plus by 40 e to the negative. Katie. So therefore, subtract 75 we get 60 is equal to 5 40 e to the negative. Katie. Um, so we get the time taken for the temperature to be 95 F as X plus one hours. So we've got Well, 95 is equal to 75. Plus by 40 uh, times e raised to the negative. X plus one, uh, times Kate. So clean this up. Subtract 75 we get 20 is equal to 5 40 Yeah, to the negative X plus one escape. Okay, let's call this, um, let's call this year crazy in one. And just hear equation, too. Yes. You know, you have no mouth so that we can divide equation one by equation two. Because they have because they have every number of energy what they don't need. Okay, so get 60/20 is equal to a while. 5 40 5 40 times e raise the negative. K t divided by by 40 times e ah, raised to the negative K plus one. I mean, I mean x plus one times. Okay. Okay. So what is good here? Um, just get well left side of this three, right? And this right hand side just reduces to eat a cake. So therefore, we got three is equal to you, the k where we just sound for K. And we get, uh, while taking natural wonder of the natural law out of both sides with the natural log of three. It's equal to, well, the natural log of e to the K. But that is just equal. Okay, so we've got 1.0 99 is equal to kick. Okay, so then, um, we can substitute the value of K in our equation one and get well, 6 60 divided by 5 40 is equal to 5 40 times e raised to the negative. 1.99 Okay, um, divided by 5 40 Those five forties was gonna cancel out, and we get that Ah, 60 or 5 40 That 0.11 is equal to you. E raised to the negative 1.0 99 x. You should be an axe, not a right. Okay, um, so then just determined value of X. We've got the natural log of 0.11 is equal to the natural log of e race to the negative 1.99 x. So we got, um, negative. Two point shoe is equal to, well, natural log of evil. Power is just the powers we've got. Negative 1.99 X and solve for X. We get X is really close to two. So get to is approximately equal to X. Okay, so therefore we get the time taken for the temperature of the object to be, Ah, 135 F as two hours after it is placed in the room. Therefore, the object be placed in the room. So the object, the object, um well, well be placed in the room. Okay? Yeah, uh, placed in the room, Um, two hours before Ah, 4 p.m. So thus, um, object placed in the room at 2 p.m. So the time that we're looking for here is two PM All right. Take care


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