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A roast turkey is taken from an oven when its temperature has reached $ 185^o F $ and is placed on a table in a room where the temperature is $ 75^o F. $(a) If the temperature of the turkey is $ 150^o F $ after half anhour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to $ 100^o F? $

a) $137^{\circ} \mathrm{F}$b) $116 \min$

03:49

Wen Z.

01:49

Amrita B.

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 8

Exponential Growth and Decay

Derivatives

Differentiation

Campbell University

Harvey Mudd College

Idaho State University

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this problem uses Newton's law of cooling, and we see when we read through the lesson in the book that Newton's law of cooling fits the model for exponential growth and decay y equals. Why not eat to the Katie? And we can also make it a little bit more problems specific if we replace the why with T minus T's of S T is the final temperature of the object. T's of S is the temperature of the surroundings, and we can replace Why not with t not minus t suggests t not is the initial temperature of the object and Tisa best is the temperature of the surroundings. So for this problem, we know the temperature of the surroundings that would be the room temperature is 75 F, and we also know the initial temperature of the object, which is turkey, because it comes out of the oven at 100 85 degrees so we can go ahead and substitute those numbers into our equation. And we have t minus 75 equals 1 85 minus 75 times E raised to the power. Katie, we haven't found k it. Okay, so what? We want to do is use this point that were given The temperature of the turkey is 150 degrees when the time is 30. We're going to use that to help us find the value of K. So we'll substitute 1 50 for the temperature, the final temperature. We can go ahead and subtract the 75 from the 1 85 and that gives us 1 10 and we can put 30 minutes. And for the time, and we'll solve this for K 1 50 minus 75 is 75. Then we're going to divide both sides by 1 10 Oops, that's supposed to say 1/10 and we end up with 75. Divided by 1 10 simplifies to be 15/22 so 15/22 equals each of the 30 K. We take the natural log of both sides and then we divide both sides by 30. So we have K equals natural log of 15/22 divided by 30. So this gives us our model, which for this problem is going to be T minus 75 equals 1 10 times E to the natural log of 15/22 times. 30/30 times T. Okay, so we're going to use that model to help us finish part A, which is to find the temperature when the time is 45 minutes. So here's that model again, and we're going to substitute 45 for the time and this all goes in the calculator very, very carefully, making sure to have all the parentheses in the right place, and we end up with t minus 75 equals approximately 62 degrees. So now we're gonna add 75 to that, and we get approximately 137 degrees, so that tells us that at 45 minutes the turkey would be 137 degrees. Okay, let's move on to part B. So in this part, we're figuring out the time when the temperature is 100 degrees so we can take the same model, which we have right here, and we can substitute 100 for Capital T, and we're gonna sell for lower case T. Okay, so let's subtract. We get 25 and then we're gonna go ahead and divide both sides by 1 10 and 25 divided by 1 10 simplifies to be 5/22 and then we're going to take the natural log on both sides. And now to get t by itself, let's multiply both sides by the reciprocal of this fraction. So we end up with t equals 30 times a natural log of 5/22 divided by the natural log of 15/22. So we also carefully put that into a calculator and we end up with approximately 116 minutes.

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