A freshly brewed cup of coffee has temperature $95^{\circ} \mathrm{C}$ in a
$20^{\circ} \mathrm{C}$ room. When its temperature is $70^{\circ} \mathrm{C},$ it is cooling at a
rate of $1^{\circ} \mathrm{C}$ per minute. When does this occur?

Answer

20.5 minutes

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 3

INVERSE FUNCTIONS

Section 4

Exponential Growth and Decay

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Video Transcript

Hey guys, welcome back for this problem. We're going to use Newton's law of cooling, which states that the change in temperature over the change in time it's legal to RK value and their temperature of the object. My temperature of the surroundings we know the temperature of the surroundings is 20 degrees C. We complain that in we have DT over de little t okay, T minus 20. Next it is said that when the temperature is 70 degrees, so when t able to 70 degrees, it is cooling at a rate of one degree per minute, so won t 0 to 70 degrees Celsius change in temperature over the change in time. People to negative one. It's getting colder at one degree permit one degree Celsius for me. So you plug those into our equation and sulfur K. We have minus one sequel to K 70 minus 20 or negative. One is evil. About 50 K or K is equal to negative point zero negative 0.2 so we can rewrite our equation. We have DT over the little T zeal to negative 0.2 temperature minus 20 and now we can actually make a substitution So if we go back up to this nation right here, let's let why be equal to t minus 20? And we can say that. Why? Prime derivative? Let's be equal to deep, right? We can actually rewrite that. The change And why change in wine over the change in t deal to K. Y. That looks more like our exponential equation that we're used to. And if you integrate both sides, you end up with wise to our initial value nice to be a zero e to the Katie. And at this point, we know that AT T is equal to zero when times equal to zero. We know that the temperature of the coffee is 95 degrees. See, we also know that why is able to t minus 20. We can say that why he was 95 minus 20 would be 75. We can say that why zero is able to 75 degrees C. That will be our initial condition. And now we're asked to figure out I need to figure out when the temperature is 70 degrees. Try and figure out won t. It's legal to 70 degrees. When does it occur? Well, of T 0 70 degrees. You know that? Why is he going to team on his 20? That's 50. Basically figure out one is why you called a 50 degrees C. So we have Why 50 Siegel to our initial. Why, friend appear was 75 degrees C 50 0 to 75 b to the K. We found Kay was 0.2 from before Temps de And now we just want us offorty. We can divide both sides by 75 50/75 equal to E to the negative points here to t We could get the natural log of both sides. You're left with Ellen. That reduces down 2/3 zero to negative 00.0 to t 40 is equal to natural log of 2/3 divided by negative 0.2 And upon doing that algebra, we could get an answer 20.5. So that's it will take 20.5 minutes. He is equal to 20.5 minutes. Cool down to 70 degrees. And that is the answer. Thanks for watching

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