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Tai-Wei (Andy) Lee May 11, 2009 IB Math SL 1 p.6 Ms. Bessette Internal Assessment II* Mathematical Modeling (Type II) The body mass index (BMI), or Quetelet index, is a statistical measurement which compares a person's weight and height. (http://en.wikipedia.org/wiki/Body_mass_index) Although it does not directly measure a person’s percentage of body fat, it is a useful tool to estimate a healthy weight according to how tall the person is. The BMI is most often used as a diagnostic tool to identify underweight, overweight and obesity. The BMI, according to the metric system and SI units, is defined as one’s weight in kilograms divided by the square of one’s height in meters. The BMI can be illustrated by the following formula: 2028825-2540 Age (yrs)BMI216.40315.70415.30515.20615.21715.40815.80916.301016.801117.501218.181318.701419.361519.881620.401720.851821.221921.602021.65 For this assignment, I was given a table of median BMI for females of different ages in the US in the 2000, the table looks like the following: Table 1: (Source: http://cdc.gov) By plotting the data given from Table 1 into LoggerPro, the graph below can be obtained Graph 1: In this graph, the variables are age and BMI; in the x-axis is age by year and in the y-axis is BMI. The parameters in this graph and Table 1 are that this data only takes in account female, in the US, and age between 2 to 20 years old. 4629150234950By observation, I noticed that the sine function best models the behavior of this graph. The reason being that Graph 1 simply looks like a sine function after it had been shifted to the right. Moreover, the graph contains an amplitude and possibly a period through my observations. Figure1: graph of sine function; y=sin(x)First of all, the general sine function is defined as y=AsinBx-C+ D where, A determines the Amplitude, B determines the Period, C determines the Horizontal translation, and D determines the Vertical translation. I then go about finding A,B,C and D to arrive to an equation (model) to fit Graph 1: To find amplitude or A, I take the maximum and minimum of the y value and subtract them, which will give me the distance between the max and min. Then I take that number and divide by 2, which will give me half the distance between the max and min, or better known as, the amplitude. How this was done is demonstrated below: Amplitude= (21.65-15.20)2=3.225 The amplitude or A is therefore 3.225 Next, to find period or B, I take the maximum and minimum of the y value and subtract them, which will give me the distance between the max and min. Then I take that number and divide by 2, which will give me half the distance between the max and min, which in this case, is half the period, so I will multiply that number by two again to get the full period in some number. Then I take that number and multiply by π180 in order to change it into radians. This final number then is my period. How this was done is demonstrated below: Period=yMax-ymin2×2×π180 =21.65-15.202×2×π180=.225 The period or B is therefore .225 Next, to find the horizontal translation or C, since I know that the general sine function passes through the point (0,0), I then simply take the x value of the median or midpoint between the peak and the trough and that number will indicate how far horizontally it has moved from the original position. How this was done is demonstrated below: Since the highest y value occurs when the x value is 20, this is the peak and since the lowest y value occurs when the x value is 5, this is the trough Horizontal Translation= 20+52 =12.5 The horizontal translation or C is therefore 12.5 Lastly, to find the vertical translation or D, since we know that the general sine function passes through the point (0,0), I then simply take the y value of the median or midpoint between the peak and the trough and that number will indicate how far vertically it has moved from the origin. How this was done is demonstrated below: Vertical Translation= 21.65+15.22 =18.425 The vertical translation or D is therefore 18.425 Finally, I arrive at an equation (model) to fit Graph 1: y=3.225 sin .225x-12.5+ 18.425 The equation can be graphically represented along with the data points from Table 1 as below Graph 2: In this graph, the variables still are age and BMI; in the x-axis is age by year and in the y-axis is BMI. The parameters in Graph 1 also hold true with this graph. In this graph, the value of C that is displayed is different from what I found because LoggerPro plots sine functions as y=AsinBx-C+ D, rather than in the form y=AsinBx-C+ D; so I just basically took .225 and multiply it to -12.5 which then gives me -2.8125 as shown in the graph. From what is displayed in Graph 2, it is justified to say that the sine function best fits with the data given from Table 1. In addition, with the help of the Ti calculator, I am able to find another function that also models the data from Table 1. Since it is already demonstrated that the sine function best models the data given, it can also be said that the cosine function will also model the data given because the sine function is the cosine function shifted to the right π2 radians; what this means is demonstrated below: π223526753810190503810 radians Figure 2: Graph of cosine function (left) compared with the graph of sine function (right); these graphs are screen-shots from the TI-SmartView program The general cosine function is defined as y=AcosBx-C+ D where, A determines the Amplitude, B determines the Period, C determines the Horizontal translation, and D determines the Vertical translation. In this case, A,B and D of this cosine function will be the same as A,B and D of sine function because they will have the same amplitude, same period and the same vertical translation; however, the horizontal translation will be different due to the reason as mentioned above, the sine function is basically the cosine function shifted to the right π2 radians. Since A,B and D are the same for the cosine function, they are: The amplitude or A is therefore 3.225 The period or B is therefore .225 The vertical translation or D is therefore 18.425 To find the horizontal translation or C, I first observe that the general cosine function starts x=0 at its y maximum or peak as shown in Figure 2. So I make the conjecture that where of x, that the maximum of y occurs is how much it has moved from x=0. How this was done is demonstrated below: x-Age (yrs)Y-BMI216.40315.70415.30515.20615.21715.40815.80916.301016.801117.501218.181318.701419.361519.881620.401720.851821.221921.602021.65 Original Table 1: Since the greatest y-value occurs at x=20, so I say that the peak has moved 20 from X=0; therefore, the horizontal translation or C, is 20. Finally, I arrive at another equation (model) to fit Graph 1: y=3.225 cos .225x-20.0+ 18.425 This equation can be graphically represented below Graph 3: In this graph, the variables still are age and BMI; in the x-axis is age by year and in the y-axis is BMI. The parameters in Graph 1 also hold true with this graph. In this graph, the value of C that is displayed is different from what I found because LoggerPro plots cosine functions as y=AcosBx-C+ D, rather than in the form y=AcosBx-C+ D; so I just basically took .225 and multiply it to -20.0 which then gives me -4.5 as shown in the graph. The sine and cosine function are both graphically represented below: Graph 4: In this graph the best fit model function, the sine function, and the cosine function are both plotted. The two models are slightly different because as said, the sine model function is the best fit model function, any other functions would just rather be a representation of the data but none can better fitted than the sine function. According to my model function: y=3.225 sin .225x-12.5+ 18.425 A 30-year-old woman in the US would have a BMI of approximately 16.68 kg/m2; how this was done is demonstrated below: Plug 29 for x because when I plug x as 1 I get, 16.7=3.225 sin .2251-12.5+ 18.425 Which is the closest to the BMI for age 2; therefore, (age – 1) = x So (30 -1) = 29 So I plug in 29 for x and I get 16.68=3.225 sin .22529-12.5+ 18.425 So a 30 year old woman in the US would have a BMI of approximately 16.68 kg/m2? That result, to many researchers of BMI, would not sound reasonable and convincing because it is hardly ever the case that a 30 year old woman would be in a better shape (BMI wise) than a 18 year old teenager. Most likely it will be that the 30 year old woman would have a higher BMI number than a 18 year old teenager. From internet, I found a set of BMI data for females from England age between 0-15 years old. The data is presented below in a form of a simplified table rather than the original gigantic chart that I found: AgeMean BMI of Girls in UK year 2002015.7116.3216.7316.6416.8516.3616.5716.8817.4917.91018.81119.91220.01320.81422.11522.2 Table 2:
(Source: http://www.erpho.org.uk/viewResource.aspx?id=8993) I then tried my model onto this new set of data shown below Graph 5: Obviously, my sine model does not fit this new set of BMI data in the UK at all. More or less, the new set of data resembles a quadratic function, so I tried it on another graph below. Graph 6: Unsurprisingly, this quadratic function works much better on the new set of data from England. In this graph, I adjusted the x-axis because in the data I obtained of English girls’ BMI, the age only ranges from 0-15 years rather than 2-20 years. As a conclusion, although the sine model I found best fits the data given in Table 1, it is not applicable to other sets of data, it is only limited to the set of data given in Table 1. One thing that differentiates the data given in Table 1 from Table 2 is that in Table 1, the numbers are medians; on the other hand, the numbers in Table 2 are means.