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Problem 15 Easy Difficulty

The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both function, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose $ t = 0 $ to correspond to 1960 or 1980.]

Answer

$P_{E}(t)=1909.7761(1.0796)^{t}+94,000$ and the logistic function
$P_{L}(t)=\frac{33,086.4394}{1+12.3428 e^{-0.1657 t}}+94,000 . P_{L}$ is a reasonably accurate

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Video Transcript

because we aren't allowed to show the actual table here. Our goal for this problem is going to be to create a model. So here's what this looks like. We want to first start by um entering data or inputting data. President. Okay, great calculator or Excel. Then you want to select data. Yeah. And create a trend line on trend line. In Excel or aggression line is what it's called the calculator. And then you want to make sure you choose the correct model um um model um form or choose the correct general form. So what I mean by that is going to be it's going to be either a linear form exponential form um Depending on what the problem gives you. And with that form you will obtain the model. And the model is really what we're after. Because once we have the model we can interpret results, predict future outcomes um or determine outcomes of unknown data points. Really that is where modeling becomes a very huge and important aspect within math and science. Um