While on Casey's computer, you notice that a model for a...

While on Casey's computer, you notice that a model for a recent experiment is completed. The model gives the temperature (T) as a function of time (t) as given below. As before, a time of t = 0 represents the time of data collection, with negative values representing earlier times. Knowing Casey, they will want to know the domain, range, and other information about the model. You review the model and provide the information that Casey will request. $T(t) = \ln(4t + 28) + 1.7$ Enter the domain in interval notation. To enter $\infty$, type infinity. The vertical asymptote is t = As t approaches the vertical asymptote, $T(t) \to$ As t approaches $\infty$, $T(t) \to$ Which of the following graphs best represents the graph of $g(x)$?

Transcript

00:01 This problem has a lot of things to do on the calculator.

00:02 First of all, i'm going to put my data into a table rather than into coordinate system.

00:07 I'm putting this data in list one, and this is time.

00:11 0, 5, 10, 15, 20, 25, and you should do the same.

00:17 And then in list two, i put in the temperatures, the actual temperatures, and then we will transform them in a second.

00:24 And this is the 78, the 66, the 57 .5, 51 .2, 46 .3, and 42 .5.

00:38 And in part a, they want us to transform those and have this be 21 degrees less than what this is to take away the room temperature.

00:50 So what the difference is between the room temperature.

00:54 You can put that into your list three and have it do the calculation of list two minus 21 and then fill in this list and i'll put in the first view.

01:04 I'm not going to put them all in.

01:06 And then they ask you to graph some things.

01:08 And you can go ahead and graph that.

01:09 But they tell you that the model is if we look at the linear regression, actually they do an exponential regression.

01:18 They don't have you do it, but they tell you that the equation comes out to be this for part b.

01:25 You're supposed to graph this data and see that it has an asymptote and graph this data and see that it has an asymptote just shifted.

01:33 So i'm not going to do any of the graphing.

01:35 I'm going to leave that for you.

01:36 I'll look at the algebra.

01:37 And then they ask you to solve for t.

01:39 And so when you add 21, you'll end up having this.

01:46 And you definitely, that's a 4.

01:48 You definitely do not want to add these, this multiplication override.

01:51 You can even put the 21 at the end, too, so you're not tempted.

01:55 All right.

01:56 Now, part c, we're doing something, doing a transformation.

01:59 They ask you to take the log of the natural log of the t minus 21.

02:07 So you would want to take the natural log of your list three.

02:12 And when you take the natural log of the list three, i'll put the first few elements down, but let me get back to my spreadsheet on my calculator.

02:20 So up here in list four, you should be putting in for list four.

02:25 You should be taking the natural log of list three, of that list.

02:32 And so we have to hit second in that.

02:34 And when we do that first element, and i'm only going to write down the first couple, 4 .04, this comes 3 .8067, and so on...

Question

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