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A cold beer initially at 35ºF warms up to 40ºF in 3 min whilesitting in a room of temperature 70 ºF. How warm will the beer beif left out for 20 min?...

Question

A cold beer initially at 35ºF warms up to 40ºF in 3 min whilesitting in a room of temperature 70 ºF. How warm will the beer beif left out for 20 min?

A cold beer initially at 35ºF warms up to 40ºF in 3 min while sitting in a room of temperature 70 ºF. How warm will the beer be if left out for 20 min?



Answers

A cold beer initially at $$35^{\circ} \mathrm{F}$$ warms up to $$40^{\circ} \mathrm{F}$$ in 3 min
while sitting in a room of temperature $$70^{\circ} \mathrm{F}$$. How warm
will the beer be if left out for 20 min?

Details that the temperature of this beer stein starts at 20 F, and it's placed in a room with the temperature of 70 F. When we're also told that after 10 minutes, the temperature of the beer stein increases to 35 F. So but we just found out waas. If we use do it laws of cooling or heating that are you not, which is the initial temperature of our object Beer Stein is 28 F. A temperature of our surrounding, which is our room, is 70 degrees. And then we also know that you of 10 is going to be 35 F. And now from that information, they want us to determine what will the temperature be of the Stein after 30 minutes. And then they also wanted to figure out how long will it take, or it to reach a temperature of 45 degrees. So this will follow in the same steps as we did with the other new in law of cooling problems and even give a hint in the previous problem to use the equation. Fortunate ones all cooling so that'll be discretion appear in the top left corner. So let's first just plug everything in that we know, and then we can see what we still need to try to figure out. So we're gonna have that u of t is equal to so t so 70 waas 28 minus 70 and then we don't know what e or we don't know what kr tr right now Since this is just in general, then we can go ahead and supply that down. So 28 minus 70 should be negative 42. So we can rewrite this as 70. Mind this 42 e to the k. T. So, if we were to try to plug in 30 into this equation news, we still don't know what Kate. So we need a sulfur cape first. And then once we do that, we can go ahead and plug in for tea. So what I'm gonna do is I'm gonna do this in general, because notice over here for our second question, we want to find the time and in solving for K, we also saw for tea in that same way. So let's do this in general first and then, But the second part, we can just plug everything into that equation we get so person when you do is get this e by itself. So we're purse going to subtract 70 on each side, so it's going to give us U of T minus 70 is equal to negative. 40 two, he to the Katie. Now we can go ahead and divide each side by negative 42. So those council over there and then we would want to take the natural log of each side's of natural log and then this natural organ that you cancel out so we'd be left with K t like this. So if we're wanting to solve for K in this instance, all we would need to do is divide over by teeth and doing that is going to tell us that K is equal to one over T natural log of U T minus 70 over negative 42 and I Let's plug in what are T and are you of TIAs? So remember, we're going to get that from this right? Here's let's just go ahead and race. What? We have some tea should be So we're gonna plug these into here, So t is 10 and then U of t are you of 10 in this case would be 35. So when we simplify that down, so 35 minus 70 is negative 35 then we would divide that by negative 42. That would give us 56 So this here is going to simplify down to tell us that K is one tent of the natural log Uh, 56 And now, with that information, we can go back up to our original equation here and plug that in. So let's just write that out. First, we have beauty is equal to 70 minus 42 and then e raised to the 1/10 natural log of 56 times teeth. And if we want to figure out after 30 minutes, all we need to do is replace this tea with 30 so about t there and then also this tea here with 30. And then once you plug old this into a calculator, it looks like we get something around 40 5.7 degrees Fahrenheit. So this is what we expect the temperature of the beer stein to be after about 30 minutes now for when the temperature is going to reach 45 F. Now, this is where us solving or over here or on putting toe Astra surround comes into help us. Because at this point, all we need to do is replace you, have tea with 45 and then also move that k over. So let's go ahead and rewrite that. Robots. So it be natural log. Oh, actually, let me. Right, Katie, On the love side, we have Katie is equal to the natural mark. Uh, you ah t minus 70 all over negative 42. And then we would want to divide each side by K. So I just put one over que in front here. But those case cancer, and then we have t is equal to that expression there. So are you of t is going to be 45. And then we also found that que was 1/10 of natural log over 56 So let's go ahead and plug all them. So we have t is equal to so be won over 1/10 natural log of 56 times the natural log of so 45 minus 70 all over minus 42. And if we were to simplify this. So if we have 1/1 10th that would just be 10. So would be 10 over natural log of 56 times the natural log of and when we simplify everything in that log there. So we have 45 minus 70 which is negative. 25. And then we divide that by negative 42 which would just give us 25/42 since the negatives would cancel out with each other. And now we can plug this into a calculator to give us what our time is. And it looks like we get 28 points. Five. And remember the units for this should be in minutes. So after about 28 a half minutes, our beer Stein will reach a temperature of 45 degrees.

Scenario were given a bottle of beer and that year is in A has a starting temperature of 28 F, and it's placed in a room with of 70 F. And Rosa, given the information that the temperature of the beer after 10 minutes has become 35 degrees, we can use a function then you in Slav cooling, which says that the temperature of an object is equal to the temperature of the surrounding medium, plus you of zero. So starting temperature minus T times E to the K Times T power or tea is a time lower. Casey's Time and K is the rates of declined temperature or, uh, growth and temperature. This case, it would be growth and temperature is its place in a warmer room. So we conserve feeling in these values were given that offer 10 minutes, the temperature has risen to 35 to 35 equal to surrounding temperature, which is 70 plus 28 minus 70 times you to the K and T. Here's 10-K times 10 and we just all be for K here, the rate of growth of temperature or increase in temperature. So there, if I have minus 70 is 35 negative for your life is equal to and 28 minus 70 is 42 negative. 42 and we can just to Sorry, this is this is Ah 42 28 minus 70 is equal to negative 42 times e to the K 10-K power and divide both sides by neck of 42. We get 35 over 42 or we can simplify that to five. 5/6 by six. He holds E to attend Kate and then take natural log about sides to get Ln a five or six sequel to 10-K Divide that by, uh, divide that by 10. Both sides do you get Ln a 5/6. Divided by 10 is equal to negative 0.18 I mean, uses value of K and the problem that asked. So what is the temperature off a sign after 30 minutes? So temperature U of t is equal to 70 plus. Now get a 42 times he to the 10 Teoh. 30 minutes of 30 times K, 30 times negative. Negative 0.18 Okay, just right cake eggs. No space. But we know. Okay, A sequel to From the previous equation Result for K so you can actually put this value Are this algebraic expression into a calculator? So 70 plus, uh, negative. 42 times E to the 30 times negative 30 times K times negative 0.18 is approximately 45.5 degrees Celsius. So this is after being the room for 30 minutes. And the question also asks how long what worlds along will take the beer to reach a temperature of 45. So now we can use 45 here. 45 is equal to get 70 plus negative 42 because soaring temperature and original, uh, and attempted the room remained the same times e to the K, which is negatives. Your appliance or 18 times teeth, this time resolving 40 about the time it takes for the beer to reach 45 degrees. So subtracts 70 from both sides. We have negative. 25 is equal to negative 42 divided by e times eats and the negative 0.1 80 I take a divide both sides by negative 42 in and take natural log. So we have natural law of 25. About about 42 is equal to negative 0.1 80 because be, uh, has cancelled out because of the natural log. And I want to divide both sides by making is your 0.18 So l n of 25 divided by 42 um, is equal to and And you are. And divide that by negative 0.18 We got approximately. Uh, this is equal to see 42 this is Eagle two 108 So negative 10 h. 70 native 101 point, uh 81 Oh, sorry. This is actually roads in the run. So this is this should be 28. Time is equal to 28. 28.28 to a 2.8 minutes. This is on the length of time it will take for the beer to reach 45 degrees Celsius.

Okay. This is Newton's law cooling problem, and Newton's Law of cooling says the rate of change of the temperature with respect to time is proportional to the difference between the body's temperature and the surrounding temperature. The temperature of the surrounding medium so air or water or, in this case, ice. So initially, the wine at time zero is 70 degrees, and then it's put into some ice. That's temperature 32 degrees, and after 15 minutes, it cools to 60 degrees. How long will it take to cool of 56 degrees? Okay, so what's gonna have toe happen is we're gonna have to use thes two bits of information here and here to find the constants after we do the integration here. And then we can answer the question. Okay. So first, this is a separable equation, so it's d t over t minus 32 equals k d little t. So little t is time big d temperature. Sorry. I forgot to say that integrate both sides get the natural log of T minus 32 equals K t plus c. All right, we're trying to solve for big T, so we need to get it out of the logarithms. So the only way to do that is to use thes as exponents for I e to the l N T minus 32 equals e to the K T plus C e and natural log are inverse functions. So they canceled. You get t minus 32 equals eat the Katie times eat the sea. Okay, See, was just a constant. So eat the sea is just a constant. So we're going to give it a new name. Let's just call it. See? It's just a different See, So we get t equals C E to the Katie plus 32. Okay, at time zero the temperature waas 70. So 70 equals C E to the zero plus 32. Okay, so that gives us that she is 40 38. So now our equation is t equals 38 e to the k T plus 32. Okay, then we know after 15 minutes, temperature is 60. So 60 equals 38 e to the k times 15 plus 32. So subtract 32 from both sides and you get 28 you to the 15 k. Divide both sides by 38 the 15 k we're trying to solve for Kay. The only way to get it down from the exponents. Take the logger rhythm of both sides believes the natural look natural log and e inverse functions so they undo each other. Alright now divide both sides by 15 to find out what K is so now. Either you can leave it like this or you can change it to a decimal three advantages of leaving it like this is you're going to get more exact. The answer. It's kind of balky too, right? So I'm gonna go ahead and convert it to a decimal. But I wouldn't have done it before now because you're gonna lose things if you dio 28 divided by 38 take the natural log of that and divide by 15. So I get K is negative 0.203 59 And I would use all of those desperate places if I were you. Just in case because you're working with big numbers and small numbers, that's just better if you keep them. So now our equation is t equals 38 e to the minus 380.20359 t plus 32. So all that work was just to get the equation ready. Now we can answer the question and the question. Waas, how long would it take to cool the 56 degrees? So we want the temperature to be 56. Yeah. Okay. Subtract 32 from both sides. Divide both sides by 38. Trying to get the tea down from the exponents. So the only way to do that take the log. A rhythm of both sides. Yellen and the evil Cancel last step. Divide by the coefficient of tea. All right. See what we get. 24. Divided by 38. Take the natural log divided by 0.20359 and I get t equals 22.57 or about 23 minutes. Does that make sense? It took 15 minutes to cool. Cool down to 60 and then takes eight more minutes because remember, it's pulling down exponentially, like e to the minus X, which looks like this. So the cooler you want it, the longer it's going to take to get there

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.


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