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