• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19

Find the median body mass index (BMI) for females in the United States of America.

Extracts from this document...

Introduction

image00.png


The table below shows the median body mass index (BMI) for females in the United States of America (USA) for the year 2000 (source www.cdc.gov).

Age (yrs)

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

Using the two variables above in the table, the independent variable (age), and the dependent variable BMI which has the units kg.m-2. The graph below was generated using Microsoft Excel 2007.

image01.png

Using a GDC, the data was fitted to a set of parameters, a screen shot of these parameters can be seen below:

image02.png

It can be seenfrom the plot that one possible mathematical fit for the graph is a sine function. This is apparent because it has the same periodic nature (it appears to be ¾ of a period of a sine curve) as a sine function. The next step in modelling a generic sine function to the data in the plot above, and applying a number of changes to f(x) = sin(x) curve. There are quite a number of calculations that are needed to acquire an accurate model of the data. Each aspect of the graph is investigated separately, for example vertical shift.

The graph below shows you the general function of a sine equation.

y=Sin(x)

image11.png

...read more.

Middle

image21.png

image22.png

Now the model becomes, f(x) = image23.pngimage23.png

image24.png

The model function image25.pngimage25.pngis now compared to the plot of the original data table (BMI).

The graph below shows the model compared to the actual data using the GDC.

image26.png

image27.png

This can also be seen using excel, the model was applied to the same independent variable values (ages), the results can be seen in the following diagram.

Age (yrs)

BMI

Model

2

16.4

15.82473

3

15.7

15.48645

4

15.3

15.27683

5

15.2

15.20501

6

15.21

15.27413

7

15.4

15.48119

8

15.8

15.81712

9

16.3

16.26725

10

16.8

16.8119

11

17.5

17.42726

12

18.18

18.08646

13

18.7

18.76066

14

19.36

19.42042

15

19.88

20.03689

16

20.4

20.58313

17

20.85

21.03527

18

21.22

21.37354

19

21.6

21.58317

20

21.65

21.65499

image29.png

As you can see there is a slight difference between the model graph and the original data graph.

They both have similar curve nature. The model function has a constant up and down curve while the original data graph has one curve and then it remains constant.

By using excel we were able to find another graph that fits the data. This is a polynomial function.

The equation for the function is X5+X4+X3 +X2+X1+X by adding your data in the graph we were able to find that the polynomial function has the best fit to the graph.

image30.png

The graph below shows the function compared to the model:

image31.png

The difference between the sin function and the polynomial function was that the sin function has constant curve movement; however the polynomial function had the best fit.

...read more.

Conclusion

Calculating the different aspects of the sine function will help to obtain best fit for the data above:

Vertical shift

image34.png

image35.png

d = 18.4

f(x)= sin(x)+18.4

Amplitude

image36.png

image37.png

 a = 3.2

f(x)=3.2sin(x)+18.4

Period

image38.png

image08.png

B =image40.pngimage40.png

F(x)=3.2sin(image17.pngimage17.pngx)+18.4

Horizontal shift

image41.png

image42.png

image43.png

 b is known to be image40.pngimage40.png

image45.pngimage45.png=12.5

-c = 12.5 x image40.pngimage40.png

-c =2.61

C= -2.61

F(x)=3.2sin(image17.pngimage17.pngx-2.61)+18.4

Using excel the model was applied to the same independent variable values (ages), the results can be seen in the following diagram.

Age (yrs)

BMI

model

2

16.5

15.82473

3

15.6

15.48645

4

15.4

15.27683

5

15.2

15.20501

6

15.4

15.27413

7

15.5

15.48119

8

15.75

15.81712

9

16.4

16.26725

10

16.8

16.8119

11

17.5

17.42726

12

18.1

18.08646

13

18.6

18.76066

14

19.3

19.42042

15

20

20.03689

16

20.4

20.58313

17

20.9

21.03527

18

21.3

21.37354

19

21.5

21.58317

20

21.6

21.65499

image46.png

The graph below shows the model compared to the actual data using the GDC.

image47.png

image48.png

-There is a slight difference between the model graph and body mass index (BMI) of young Australian woman. They have a similar curve nature for a specific period before the sine function changes.

-The sine function has a constant up and down curve while the original data graph has one curve and then it remains constant.

-The BMI of the Australian females is very similar to the BMI of US females from the age of 2 -20.

-Until this point the sine function has been able to calculate the BMI of different woman in different countries, however if you apply more tests to the function with different countries and ages then you will see slight limitations to the graph which would require you to either recalculate the equation or use other functions.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. This portfolio is an investigation into how the median Body Mass Index of a ...

    If this data was a record of the extremes rather than the median BMI of a population, the y values could lower than the window setting of 12 for this graph, as well as much higher than the window setting of 24.

  2. Derivative of Sine Functions

    Say a=2 g(x)=2sinx , g�(x)=2cosx (graph is shown below) When a<0 the curve will be reflection along x-axis and then vertical stretch by a factor of a Say a=-3 g(x)= -3sinx ,g�(x)= -3cosx (graph is shown below) When a=0 g(x)= 0 , g�(x)= 0 the curve is still exist as y=0.(graph is shown below)

  1. Investigating Sin Functions

    Now, what happens when the B value is negative? *for above graph* Black is y=sin(-x), Red is y=sin(-2x), Blue is y=sin(-3x) As we observe the graph, we can see that the negative symbol has reflected the graph across the y axis.

  2. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    So far in this investigation we have been gathering the shape and look of the graph from just the equation, however we will now do it the other way around. We will find the equation of the curve from the graph.

  1. A logistic model

    The interval of calculation is 1 year. Please note that from year 6 and onwards the population resonates above and below the limit (yet due to the rounding up of numbers this is not observed), and it finally stabilizes in year 17 (and onwards) where the population is exactly equal to the sustainable limit.

  2. Population trends. The aim of this investigation is to find out more about different ...

    The number which represents is the only point which is exactly on the line, the year 1950 is represented as being the one on the and at this point in time there was a population in China of 554.8 million people.

  1. Investigating the Graphs of Sine Function.

    It can be noticed that B affects the period of the wave. When varying B, it can be noticed that the wave's period is B times as short as the base wave's period. In other words, the inverse of B (1/B)

  2. Portfolio: Body Mass Index

    Using the graphical calculator I calculated the BMI of a 2-year-old girl. This is what I got: The answer (16.398) is very close to real BMI of a 2-year-old girl, which is 16.40. This little testing gives me the courage to proceed with the task and to estimate the BMI of a 30-year-old woman in the USA using my model.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work