on-a-cold-winter-day-left10circ-mathrmfright-an-object-is-brought-outside-from-a-70circ-mathrmf-room

Name: on a cold winter day left10circ mathrmfright an object is brought outside from a 70circ mathrmf room
Uploaded: 2020-06-16
Duration: 5 min 43 s
Channel: Numerade
Description: On a cold winter day $\left(10^{\circ} \mathrm{F}\right),$ an object is brought outside from a $70^{\circ} \mathrm{F}$ room. If it takes 40 minutes for the object to cool from $70^{\circ} \mathrm{F}$ to $30^{\circ} \mathrm{F}$, did it take more or less than 20 minutes for the object to reach $50^{\circ} \mathrm{F}$ ? Use Newton's law of cooling to explain your answer.

Question

On a cold winter day $\left(10^{\circ} \mathrm{F}\right),$ an object is brought outside from a $70^{\circ} \mathrm{F}$ room. If it takes 40 minutes for the object to cool from $70^{\circ} \mathrm{F}$ to $30^{\circ} \mathrm{F}$, did it take more or less than 20 minutes for the object to reach $50^{\circ} \mathrm{F}$ ? Use Newton's law of cooling to explain your answer.

Michael Jacobsen · Accepted Answer

in this problem. We&#x27;re told that T at T is the temperature at time t in minutes after team minutes, it&#x27;s been removed from an up a room. We&#x27;d like to know in this problem what the temperature will be 20 minutes later. And for this problem, using the formula for Newton&#x27;s law of cooling temperature at Time T is equal to the surrounding temperature. T sabb em my a c times e to the negative. Katie. So what we need to do next is hold off on evaluating t 20 and first go through all the initial conditions that have been given. Our objective is to determine what the constant Tisa BM is C as well as K or to determine Tisa. Bam. We can just do an immediate substitution that t sabb em is equal to 10 F will obtain temperature at Time T is equal to 10 minus See Time&#x27;s each of the negative. Katie, the next constant that we&#x27;d like to determine is see itself. We can do that from the very first initial condition. The starting temperature or th zero is 70 degrees. Let&#x27;s use this last differential equation solution that we obtained to make that substitution. Well, right. T s zero is equal to first. It&#x27;s 10 minus C times e to the negative K time zero. But we also know that each of the zero result in one. So this factor cancels out immediately. The next substitution will make is that TF zero itself is 70 degrees. So now we have an equation. 70 equals 10 minus c. Subtracting 10 from both sides obtains 60 is equal to negative c So negative 60 degrees itself is that constant? C Let&#x27;s and put that back into the differential equation. Now we have two negatives to deal with. So temperature at time T will be 10 with a positive 60 times E to the negative. Katie, we have one last differential equation. Initial condition to consider TF 40 is 30 degrees or the temperature 40 minutes later decreased to 30 degrees, and that could be used to solve four K. So as we did before, let&#x27;s use this last step of this equation. We&#x27;ve been building a substitute in 40 for Time T. If we do that, then T at 40 will be equal to 10 plus 60 times e to the negative of 40 times k. Ah Further substitution tells us that T A 40 itself is 30 degrees, so that equation becomes 30 equals 10 plus 60 times each of the power negative 40 k We&#x27;ll solve this equation for K to do that vs subtract 10 from both sides we obtain 20 is 60 times each of the negative 40 k If we divide 60 into both sides will get to six after reduction or 1/3. So now we have 1/3 equals e to the power of negative 40 k and this can be solved for K by first taking the natural logger them of both sides. We obtained that natural auger them of 1/3 is equal to negative 40 k So finally, after dividing both sides by negative 40 que is now equal to negative 1/40 times the natural log of 1/3 and that will be substituted back into this portion off that differential equation solution. We just have to be mindful of the double negatives. So our next step is that temperature at Time T is equal to 10 plus 60 times the base exponential to the power of 1 48th times the natural log of 1/3 times t. Finally, we can address what is the temperature 20 minutes after this up, object has been removed from the room we can write Temperature at 20 minutes later is equal to buy substitution from the last step of our equation. 10 plus 60 each of the power of 1/40 times the natural log of 1/3 times 20 for a place of tea. If we put this all into, a calculator will obtain that this is approximately equal to 44.64 degrees, and then what this tells us is it takes less than 20 minutes for the temperature to decrease to 50 F.

Ashima Tiwari · Answer

Okay, So noticed all this question We need to use the New England cooling. Um, So what we&#x27;re doing here is we need to solve for X. Since that will give us the minutes it takes to get to t X peopling 65. So let&#x27;s start with plugging in the values we know. So we know that we want the temperature tend of Equalling 65. Um, the surrounding temperature is 75. So bold Soon the initial temperatures and it&#x27;s taken out. Um, that&#x27;s seeking Out of it is 35 again, minus 75. And then we have he toothy negative. Zero point 031 times x. So we want to solve for this X right here. So what we can start doing is by subtracting 75 by both sides. Oh, that&#x27;ll give us negative 10 on this side. We can even do 35 minus 75 to simplify this side, which is negative. 40. And then you can eat the negatives. Your point. See her 31 Thanks. And now what we want to do is, um, by both sides by negative 40 to isolate this e um, so we get 1/4 can be the negative 0.31 fax. So now, in order to isolate, um, this guy over here, we basically want to l and both sides, because when we Ln this the e basically canceled out since it&#x27;s Helen E to the e. So what we will get is Ellen one for it cools negative. Zero 0.0 31 x So let&#x27;s just people again. L enough. One worth real quick. So that&#x27;s around negative. 1.38 feet six. And now we can just But I hear there divide pulls sides by negative 0.31 that we can isolate that X and then we get exp as 44 0.719 minutes and that will be your answer. That will be the amount of time it takes to get the temperature to 65 degrees.

Lourence Gonhovi · Answer

all right. In this case, we&#x27;re supposed to find out how many minutes with the temperature will be foot five degrees. Foreigners. Ah, the fists processes to a model. The he is toe model the function that defies the cooling off the steak. So it&#x27;s given by the Newton&#x27;s law of cooling which safeties it was to see t my last Tuesday. Oh, my Nazi eight of the park. It&#x27;s here and we are given that after 10 minutes, the temporary child will be though their prison to steady decrease that degrees and the room temperature ISS 65 degrees. And the initial temperature is 24 degrees and violence that room temperature then e to the part. This is up in Dr 10 Minutes time skate. Then from there we can jump on necessary steps. Just we have to move everything from the all right inside of the left. First Liberty, subjecting 60 85 from Teddy Bailey divide. I make it you put one to get aids as there a point 854 which will then be equal to 10 to the party. Then we find natural logarithms both sides and we find 10 K. If you go to Ln 0.854 then divide by 10 and then we find that K is equal to negative 0.158 Then we can put our model complete its tease. It goes to 65 minus what d five you to the power off. Negative zero points. Zero on 58 See? And the question requires us to find after Roman minutes with the temperature before departing, please. So we are now using all my audio temperature is 45 declares 65 miners. This is not football, but for 12 minus 4 to 1 e to the appointment of unsettled plains little on 58 t. And so we subject negative 4 to 5 on the left and then divide by negative foot one. And then we get zero points for 88. Cheesy goes to e to the Boer negative 0.158 teeth. And then we find natural logs both sides, uh, and and the negative zero points Little 15 80. Then this country routes and then we are left with the negative 0.1 58 Easy goes to an end. Settle point for 88 Divide by they get exurban 0158 Both sides then get that T is equal to Ah, full 0.5 for one minutes.

Taylor Shimono · Answer

So from this problem on the most important information for us to pull out is a temperature at T subzero temperature, T&#x27;s of 10 and the temperature of the surroundings, which is 80. So when we substitute the of this information into your Newton&#x27;s law of cooling cooling, we get a d T. D. T is equal to k times the temperature of the object minus the temperature of their surroundings, which is 80. And so from here we can go ahead and separate are variables. And when we do that, um, we&#x27;re going to get the left side of her equation. Um, will be won over T minus 80 times D capital teeth is equal to okay, times de Lorca&#x27;s teeth. And so, from here, we can go ahead and integrate both sides of our equation. And when we do that, we get the natural log of T minus 80 is equal thio Okay, T plus C on this teeth, the lower case T stands for time. And so from here we can go ahead and solve for ah, the temperature of our object, which is our capital teeth as a function of time. Um and so when we do that? Um, I&#x27;ve gone and done it already. We are going to get, um the temperature of our object is equal to a C to the K t power plus 80. And so from here, we can use our initial condition to solve. Uh, this plus 80 is not in the explain it, by the way, so we can use our initial condition to solve for our a variable. So our initial condition is the temperature is 10 at T subzero. And so when we solve, we get a is equal to negative 70 and we&#x27;re going to next. Go ahead and use our second condition. Um, which is at T&#x27;s of 10 our temperatures equal to 20. And this will be equal to negative 70 e to the K times, uh, 10 power plus 80. And so when we solve, we get K is equal to negative 0.15 So now we have our equation completely determined, which is? The temperature over of our ice cream is going to be equal to negative 70 e to the negative 0.15 t power plus 80 and so to solve for the time when our ice cream is going to be a 40 degrees Fahrenheit. All we have to do is plug in. T is equal to 40 and we or excuse me, the big T temperature is equal to 40. And when we do that, we find that our temperature will melt. After approximately 36 minutes, it will be 36.303 minutes. And so this is actually going to be 26.303 minutes from the last time we measured after 10 minutes.

Khanh Ha · Answer

Hey, guys. So we have this question right here related to the Newton&#x27;s law of cooling. So a frozen steak with an initial temperature of 28 F was left too tall in a room with a temperature of 75 to please Fahrenheit. All right, so we know that the initial temperature was 28. Right? So that&#x27;s 40 0 and the temperature of the environment of the room 75. So does foresee. Okay, you got this. All right. So what we don&#x27;t know here is K, and we can solve it by using this information. So after 10 minutes, the temperature of the stick has rising to 38 degrees Fahrenheit. So the new temperature is 38 right? And T s 10. Awesome. So let&#x27;s try to figure out what K is to complete this model. So first, let&#x27;s subtract 75 from both sides. Right? Then we&#x27;ll have so training. Eight minus 75 is negative. 47 right here. All right, so 38 minus 75. This negative 37. Right. Okay, so then last divide both sides by negative 47 just like this. And then let&#x27;s take natural lock on both sides so that we can eyes elite our excellent end of the year. So then we&#x27;ll have to see quaking right here. Okay, So finally to software k last divi, but size by 10. All right, so let&#x27;s put this into a calculator and figure out 10 shall we? Right? Do you just sat here, so Okay. Should be approximately negative. 0.24 All right, so our model should be key equals CS 75 minus 47. Okay. To the power of negative 0.0 to 5 to four. Small tea chills like this, right? And now let&#x27;s get down to the actual question here. So, after how many minutes will the temperature of the steak be 50 F? Sold? The new temperature in this case is 50 right? And we&#x27;re looking for small T, which is a temperature, which is how long the state rule rich the temperature. Okay, so let&#x27;s try to solve for this tea right here. So very similar to what we just did. Less a truck boat size by 75. Alright. So minus 75 to both sides. We&#x27;ll have this right here. Okay. Very similar to what? We just did it last divide both sides by 47. So because both have negative signs. So that&#x27;s why I says Muzzi. Negative signs here and then to isolate exploded again. Let&#x27;s take the national lock on both sides. So then we&#x27;ll have to see question right here and then to calculate t. So how long will it take? Thes sexual ridge 50 F would divide both sides by negative 0.24 So last office, send out copulate er and see what will happen. Right? So then our tea should be approximately 26.3 minutes, but there you go. That&#x27;s how we saw this question. Well, thank you so much for listening until next time.

Ankit Gupta · Answer

so I have a question given to me Over here it is regarding heating or cooling off body. So here he is, the temperature of the body, given at any time. T denotes is a surrounding temperature, and the one is a body temperature. Now the question is a cup of coffee with temperature 105 degree for tonight. So here I&#x27;ll assume that do you want is a call. 205 played in a freezer temperature zero degrees for tonight&#x27;s OT. Nor did he called a zero. I&#x27;ll tell so time these mentioned to me as five minutes, so I looked up. Later, Formula on DA. I can see that the temperature of the coffee is 70 degree for tonight&#x27;s tea at tea is equal to 70 degree foreigners. So using this formula, I just don&#x27;t get 70 is equal to Do you know this zero any of us, plus the 105 minus zero into irritably power minus K in tow. Five. So from here I&#x27;ll get that seventies equal toe 105 into the rest to the bar minus five key. Now they&#x27;re asking, or Desert template, sir, after 10 minutes. So Deaton value will be, what, zero plus 105. So they the 705 minus zero indoor era to the power minus 2010 k entered off is equal to five. This time I&#x27;m going to put pizzicato 10 So cable demand as it is but instead of it will come us. So that is nothing but 100 and five into. Here&#x27;s to the power minus five k. A whole square. Can I say like that? He is absolutely I can say like that. So if I call this the question as a question number one so this valley will come out to be 105 into He was two for minus five. Gave earlier I can substitute from here. That is nothing but 70 divided by 105 Hold square the times after shooting. From where? From the question number one. So this on simplification will come out to be 17 70 that is 49 double zero divided by we&#x27;re not fight so the nice after toward these values I&#x27;ll get it like this one comes out to be 46 point 67 Big Guerry for tonight. So that is going to be my answer After 10 minutes. I hope it&#x27;s clear. Thank you

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Find the equation of the orthogonal trajectories …

Problem 10 Medium Difficulty

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$44.64^{\circ}$
Less than 20 minutes
decrease to $50^{\circ}$

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$44.64^{\circ}$Less than 20 minutesdecrease to $50^{\circ}$

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decrease to $50^{\circ}$

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