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Find an exponential function determined by the data in Table 5 .$$\begin{array}{|l|l|l|l|}\hline 5 & 10 & 15 & 20 \\\hline 12 & 15.6 & 20.28 & 26.364 \\\hline\end{array}$$

$$9.23077(1.3)^{x / 5} \approx 9.23077(1.053874)^{x}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 2

Exponential Functions

Campbell University

Oregon State University

Harvey Mudd College

Lectures

02:51

Find an exponential functi…

03:25

03:05

00:35

Find exponential functions…

00:32

00:39

00:29

01:06

01:17

Find the exponential model…

00:26

Enter the data from each t…

01:03

For the following exercise…

01:01

So if we want to find an exponential to do this, um, you could just really use any two of these points. It doesn't matter which ones you use. I'm just gonna use the first two. But if you use one of the other pairs, then you might get a slightly different answer. But I'm just going to use those to to start. So what I'm going to first do is construct two equations from this so that first one is going to be 12 is equal to a B to the fifth, then construct a second equation where it's 15.6 is equal to a times B to the 10th. So I'm just plugging those points into this equation over here. Now, what I want to do is divide each of these equations I'm going to divide on the other side, notice how the A's council out and we're going to end up with 15 point 6/12 is equal to will be to the 10th, divided by B to the fifth. It is going to be beat the fifth, and then we can take the fifth root on either side of this, um or just to the 1/5 power as well. And then that is going to give us B is equal to 15 point 6/12. Raise to the one. Actually, let me see what 15 point 6/12 simplifies down to. So that's 1.3. So we get B is even to 1.3 raised to the 1/5. So now to solve for a we could just take this, plug it into either of those equations there, um, and then solve, so I'll just plug it into the first one. So we have 12 is equal to a times 1.3 raised to the fifth raised to race to the 1/5 and then race to the fifth. So that would just be 1.3. So that would give us a is equal to or I should say, that was cancelled. So then we just get 12/1 0.3 and 12, divided by 1.3. Um, it's just 1 20/13. So we have a 1 20/13. So our exponential, or at least one possible exponential that could be used to represent this data is going to be 1 20/13 times, 1.3 raised to the 1/5 all raised to the X. And so again, if you use the other points, you might get a slightly different answer. Um, I'm not too sure because I didn't check to see what those other ones were, but at least this is one possible.

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