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# Body Mass Index

Extracts from this document...

Introduction

Body Mass Index

SL Math Internal Assessment Type 2

Body mass index (BMI) is a measure of one’s body fat. BMI is calculated by taking one’s weight (kg) and dividing the square of one’s height (m). BMI does not directly measure a person’s percentage of body fat but is used to determine of a person is underweight, normal, overweight or obese. BMI can be shown through the following formula:

The table below gives the median (the average, found by arranging the values in order and then selecting the one in the middle). BMI for females of different ages in the United States in the year 2000.

Table 1:

 Age (years) 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)

Using a TI-84 calculator (or any other graphing tool) one can set up a list where list #1 is age (represented on the x-axis) and list #2 is the BMI (represented on the y-axis). Below is a screen shot of the ages and their respective BMIs.

Then the screen shows the points plotted on the graph. Below is a screen shot of the plotted points:

Graph 1:

This graph represents the BMIs of females in the US between the ages of 2 and 20.

Middle

Next, in order to find the period (B) of this function we must again subtract the y-minimum from the y-maximum and divide by 2. This will give us the amplitude (as well as half the period), which we then have to multiply by 2 in order to find the full period. The period must be in radian, however, so we must again multiply the number but this time by . This final number is then the period of the function. The calculations are represented below:

→

Therefore, the period or B for this function is 0.225.

Next, to find the horizontal translation (C) of the function one must find the peak and trough of the function. The maximum y-value of the function occurs at x=20, the peak is therefore 20. The minimum y-value of the function occurs at x=5, so the trough is 5. In order to find the horizontal translation we must add the peak and trough values and divide by 2. The number that is produced will indicate how far the function has moved from its original position, which is at (0,0).

→

The horizontal translation is therefore 12.5.

Lastly, in order to find the vertical translation (D)

Conclusion

From the internet I found statistics regarding the BMI of women in the United Kingdom from the ages of 0-15 in 2002.

Table 2

 Age (years) BMI 0 15.7 1 16.3 2 16.7 3 16.6 4 16.8 5 16.3 6 16.5 7 16.8 8 17.4 9 17.9 10 18.8 11 19.9 12 20.0 13 20.8 14 22.1 15 22.2

(Source: http://www.erpho.org.uk/viewResource.aspx?id=9001)

Graph 5

These plots do not correspond with the curvature of the sine function used for the data in Table 1.  Also, the data in Table 2 does not fit the original equation (model) as the data only ranges from 0-15 and not from 2-20 (this required an adjustment of the x-axis). Another limitation of the model is that the data in Table 1 are BMI medians and the data in Table 2 are BMI means. Therefore, the original equation, , does not apply to all sets of BMI data but has to be adjusted in order to fit specific data.

[1] Amplitude: the distance between the maximum and minimum y-values divided by 2

[2] Period: the interval from one repetition to the next

[3] Horizontal translation: a movement of the graph along the x-axis

[4] Vertical translation: movement of the graph along the y-axis

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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