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The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form.An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at $65^{\circ} \mathrm{F}$. After 10 min the beam had warmed to $35^{\circ} \mathrm{F},$ and after another 10 min its temperature was $50^{\circ} \mathrm{F}$ Use Newton's Law of Cooling to estimate the beam's initial temperature.

$5^o F$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

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So we have this beam that we go and place outside, with the temperature being 65 F. So what they tell us is, after 10 minutes, the beam has been warm to 35 F and another 10 minutes which would mean 20 minutes later it was 50. What we want to do is determine what is the initial temperature of the beam. Using new ins law cooling, hurry snow overhead. The top left corner we could think about this is we'll get two different solutions to work with or at least we have to initial conditions to work with. So our initial temperature is just a tch not are outside is going to be HS. It's very different colors. So this is a TSH sub s. And then or each of our equations, this is going to be a TSH, and that's point to be H. So the 35 is a church for the 1st 1 and it is a tch for the 2nd 1 There a 50 and in our time for the 2nd 1 is 20 minutes in our time for the 1st 1 is 10 minutes. So let's start with the red or the first equation and see what we get when we plug everything. It so h its final temperature should be 35 minus its surrounding which we were told. It's 65 and this is going to be equal to our initial, which we don't know yet and then minus the surrounding, which again is 35 and then e to the negative k times 10. So let's simplify the left side there, and that is going to give negative 30 is equal to whatever we have on the right section. Let me just write that above so I don't have to write another line, so it'll just be negative 30. Now let's use our second equation now and plug everything in. So that's going to be so on the left hand side. It's h bias. So in this case, H is 50 are surrounding is still 65. So we have a tch not minus 65. And this would be easy to the naked do k times 20 this time. And so on the left side. This should give us negative 15. Now. Now, notice that in for both of these, we have this h not minus 65 turn. So what I'm going to do is divide equation one by equation too. So we're going to divide each side like this and notice how the h not minus 60 five's cancel out with each other, So we're going to end up with so Equation one over equation too, is so negative. 30 divided by negative. 15 is too. Like I was saying, the h not minus 60 five's cancel out with each other and then we would have e to the negative Kate. Bye bye e to the negative k times 20 or eat a negative K Times 10 divided by E to the negative K times 20 which would just give us e juvie. 10 K. Now we just need to solve the case. So we take the natural log on each side so we get natural lava tubes, is able to 10 k and then we would want to divide like 10. It's okay, is going to be too natural log of to over 10. So that's going to be our value for Kay. But we don't really care about that since ah, our answer. But we can now use this value for Kay in either of our 1st 2 equations to help us solve for H Not so Let's just use of the 2nd 1 here. So it's supposed to be negative. 15 is equal to each, not minus 65 e to the negative. And I'm just gonna leave it as Katie for right now, so we don't have to plug in that fraction right away. And actually, that should have been 20 up top. Richard, let's plug it in because it looks like it actually simplify its down a little bit now that I'm looking at it. So it's natural Log up to over 10 and then times 20. And remember, this is just from plugging in K into our second equation. So 10 divided by 20 is to as negative, too. So, actually, I was gonna write this up here to kind of show how this simplifies, so we'd have negative to natural log of two, and we can pull that negative two in as a power, which is going to give us e to the natural log of 1/4 and the eat and natural all cats out. So that would just give us 14 So this is gonna be one for so we're gonna have Negative 15 is equal to H not minus 65 times one for so we can multiply each side by four to get negative. 60 is eager to h dot minus 65 then we would want to add 65 on each side. You get by visit to H not so is going to be 5 F or our initial.

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