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IB Coursework Maths SL BMI

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Introduction

Lydia Smith

Body Mass Index

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

Introduction

In this assessment, I will be modelling the data provided in the table below, which represents the median BMI for females of different ages in the US in the year 2000. I will focus particularly on the different functions which could be matched to this data, as well as comparing this data to similar data showing women of another country.

The table below shows the aforementioned data. Age is found in the ‘X’ column because this is the independent, constant variable. The BMI values are found in the ‘Y’ column because this information is the dependent variable.

 Age (years) BMI 2 16.4 3 15.7 4 15.3 5 15.2 6 15.21 7 15.4 8 15.8 9 16.3 10 16.8 11 17.5 12 18.18 13 18.7 14 19.36 15 19.88 16 20.4 17 20.85 18 21.22 19 21.6 20 21.65

The graph below is a representation of the previous table. The graph was plotted using Autograph.   The independent variable for the above graph is the age in years, and this is shown on the x-axis.

The dependent variable is the BMI, and this is shown on the y-axis.

The highest BMI is 21.

Middle

20.85 = 289a + 17b + c

I can use simultaneous equations to find the values of a, b and c. First, I will substitute the first equation I found (line of symmetry equation) into the second equation.

15.7 = 9a + 3(-10a) + c

15.7 = -21a + c

Next I will substitute the first equation into the third equation.

20.85 = 289a + 17(-10a) + c

20.85 = 289a – 170a + c

20.85 = 119a + c

To use simultaneous equations:

20.85 = 119a + c

-

15.7 = -21a + c

=

5.15 = 140a

In order to find a:

a= = 0.0368 (to 3 s.f.)

If we substitute this equation into the first equation:

b = -10a = -10(0.0368) = 0.368

Therefore, if we substitute this to the full equation:

20.85 = 119(0.0368) + c

c = 16.47

Therefore, the quadratic equation is:

y = 0.368x2 – 0.368x + 16.47

Using information from the equation y= 0.368x2 – 0.368x + 16.47, I put all the new y values in a table (shown below). To compare the

Conclusion

Conclusion

While I did manage to find some functions that resembled the original equation, they were all deeply flawed. As previously mentioned, in the polynomial quadratic function equation, the shape of the plotted points are close to accurate, but the trend for the rest of the line does not fit at all. This makes using this function to predict future results impossible. In the cubic polynomial function equation, the shape of the plotted points are even closer to accurate than the polynomial quadratic function equation, and the trend for the rest of the line is much closer to that of the original equation, but still the line does not fit. The quartic function modelled the behaviour of the graph best, but it is only accurate and therefore reasonable up to the age of 20. We see  that after reaching 20 on the x-axes, the graph descends to the point where the BMI would be completely impossible for that age group.

I tried many other functions, but could find none that were closer and therefore none that solved this problem.

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

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