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IB SL Math Portfolio- Body Mass Index

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Introduction


SL PORTFOLIO TYPE II
“Body Mass Index”


BODY MASS INDEX                                

Throughout this portfolio, various functions will be evaluated, applying the given data.
A model function will be determined and extrapolated as it relates to the following real-world example:

Body mass index (BMI) is a measure of one’s body fat. It is calculated by taking one’s weight (kg) and dividing it by the square of one’s height (m).

The table below provides the median BMI for females of varying ages in the US in the year 2000.

Age (yrs)

BMI

2

16.40

3

15.70

4

15.30

5

15.20

6

15.21

7

15.40

8

15.80

9

16.30

10

16.80

11

17.50

12

18.18

13

18.70

14

19.36

15

19.88

16

20.40

17

20.85

18

21.22

19

21.60

20

21.65

(Source: http://www.cdc.gov)

When graphed:

image00.png

(The independent variable being the age of the women studied <x>, and the dependent being the BMI of these women <y>. Both values must always be greater than zero. )

This graph’s behavior is most nearly modeled by the cosine function,image01.pngimage01.png, because it is undulating and periodic: it repeats a pattern as it rises and falls. However, due to the limitations presented by the nature of the given information itself, only the portion of the

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Middle

through trial-and-error, and decided that the function
f(x) =3cos(.2x-4)+18most closely modeled the data, though certain values did not perfectly correlate. Below is a graph showing both the given BMI data and this model cosine function, for comparative purposes. It can be seen that the actual raw data very nearly coincides with the model: they curve in a relatively similar fashion, for example. The graphs are not, however, identical. Certain points do not exactly line up with this model. The first few, for instance, before age five, have slightly higher x-values than the cosine model, as do the last few, after age sixteen. Other values also do not correlate perfectly. Nonetheless, the model illustrates the provided BMI statistics with relative accuracy.

image04.png

In order to discover a different function that models the given information, I used technology.  I downloaded Graph 4.3, and saved the given data, the cosine function, and the model function into the program. When I compared the graph of the BMI statistics with my own model, I noticed that they did not look as

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Conclusion

 to be a bit of a closer match. Below is the data graphed with this new model function, for comparative purposes.

image02.png

It can be observed that this model does not perfectly correlate with the data, but does reflect its trends with relative accuracy.  Because the data for Guatemalan women was more sporadic than the data for American women, it was much more difficult to discover a model that fit. In other words, the data did not always follow a consistent undulating pattern, while the first set did, making it harder for a function to correspond. There was also less data present, as there is a gap between ages seven and eleven and it stops at age eighteen, which also made it difficult to view any trends. The data often shows slightly higher x-values than the model, and at some points, they are lower.  Because I used a trial-and-error method (on both a GDC and the program Graph 4.3) to discover it, the model function I found may have somewhat limited accuracy, though it does illustrate the trends this data presents pretty closely. The function, while not a perfect match, is appropriate considering the limitations presented by the data itself.


[1]http://www.cdc.gov/healthyweight/assessing/bmi/adult_bmi/index.html

[2]http://www.hughston.com/hha/a_15_2_4.htm

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