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The population of the United States from 1815 to 1975 is given in Table 9 . Determine an exponential regression function that will best fit this data. (Let 1815 be year 0, 1825 year 1 and so on.) Based on your solution, estimate the population in 2005.$$\begin{array}{|l|c|}\hline \text { Year } & \text { Population (in Millions) } \\\hline 1815 & 8.3 \\1825 & 11.0 \\1835 & 14.7 \\1845 & 19.7 \\1855 & 26.7 \\1865 & 35.2 \\1875 & 44.4 \\1885 & 55.9 \\1895 & 68.9 \\1905 & 83.2 \\1915 & 98.8 \\1925 & 114.2 \\1935 & 127.1 \\1945 & 140.1 \\1955 & 164.0 \\1965 & 190.9 \\1975 & 214.3 \\\hline\end{array}$$

$$11.147747(1.222709)^{x}, 508.541 \text { million }$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 2

Exponential Functions

McMaster University

Baylor University

Idaho State University

Lectures

01:45

The table gives the popula…

02:15

Us population The table gi…

02:09

01:27

04:38

The table gives the US pop…

06:14

02:45

06:26

03:34

Use the exponential growth…

Use the data in the table …

08:10

So if we want to come up with an exponential or this data here, what I'm going to do is just use Google sheets to do this. But if you have some other kind of calculator, um, that will do this. You can use as well Microsoft Excel will work as well. Um, but since this is probably the easiest thing I would say most people have access to, I'll just show you how to do it on here. So I went ahead and got the data, and I just wrote it in here. And then what I'm going to do first is they said they wanted us two start where 18. 15 is a time of zero. So I'm just gonna put a column for T here, and what I'm going to do now is do equals. Click on the cell to the left and then do minus 18 15, and then I guess it's gonna suggest this. But if this doesn't come up, just take this and then scroll all the way down, and then it will subtract off everything. Okay, um and now we have that filled out and we can do now is highlight These two rows here or columns, and then I'll come up to here. Where is his insert chart? And then it should plot it for me. So this is just what that data looks like. So we don't want a line or whatever this is. We wanted an exponential. So what we'll need to do is come over here to customize, um, do series scroll down a little bit and we're going to add a trend light. Now, we don't want to line. We want an exponential, so we'll click on that. So that light blue equation is supposed to be our, um, approximation of the data, and we're going to come down here to label you use equation, and then this is the exponential that they end up giving us. Um, so this is 15.5 he raised to the 0.1 seven X. Um, so depending on what calculator you use, it may give you something slightly different from this just due to like rounding errors and different kind of algorithms they may use to get it. Um, but this is at least what Google Sheets is telling me. Um, now the next thing they said is they want us to find What is the year in 2005? Uh, make sure it's 2000 and five before I say that. Uh, yeah, 2000 and five. So all we need to do is plug not 2000 and five, But let me actually figure this out over here. So if I have 2005, um then remember, we're going to take this and do Subtract 18, that one here, over. So click on it, minus 18. 15. So that gives us 194. That was actually a bad place to pull it. Let me just put over here on the side, Calvin. I've and then equals to this 18. 15. So 1 90. So you want to take 1 90 plug it into our equation here? So I'll just do that over here, so I'll do equals again. So the 15.5 times, uh and I don't think it just takes e, so we'll have to do exponential e x p like this, the new brackets and then zero point 017 times of 1 90 then just close that hit. Enter and then this should be our approximate population in millions of people. And so again, depending on what your calculator is saying is you're correct equation, um, you might have something different. I mean, it could be vastly off. I think the one in the book says it's around like, 500 million, as opposed to what we have here. Um, but I would say that's okay. It's just depending on, like, what algorithm was used to spit out your answer? Yeah. So, again, this up here is going to be our equation. At least the one that Google Sheets, um, says we should have. And then down here is our approximate population around 2000 and five.

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