- Level: International Baccalaureate
- Subject: Maths
- Word count: 1843
Maths BMI
Extracts from this document...
Introduction
BMI Internal Assessment
By: Jason Chau
Year 11 SL Mathematics - 2009-08-09
Year 11 SL Mathematics – 2009 |
BMI Internal Assessment |
By: Jason Chau |
Introduction
This piece of work is based on the data of the median BMI for females of different ages in the US in the year 2000. I will be modelling the data using different types of functions, as it is a modelling task.
The Body Mass Index or BMI is a ratio of a person’s high in relation to their weight. It is calculated by dividing the person’s weight (kg) by the square of their height (m). This ration is usually used to determine a person’s health.
Ages (yrs) | BMI |
2 | 16.4 |
3 | 15.7 |
4 | 15.3 |
5 | 15.2 |
6 | 15.21 |
7 | 154 |
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 technology, plot the data points on a graph. Define all variables used and state any parameters clearly.
I put the age in the x column because it is the constant variable. I called this column the ‘Age’ column
Next, I put the BMI values in the y column as it is the dependent variable and called it the ‘BMI’ column
The graph is plotted using these data points in the table with the all points unconnected, as this makes the graph more accurate.
Middle
20.85 = 289a + 17b + c
Now that I have 3 equations, I can use simultaneous equations to solve the values of a, b and c.
Substitute (1) into (2)
- 15.7 = 9a + 3(-10a) + c
15.7 = -21a + c
Substitute (1) into (3)
- 20.85 = 289a + 17(-10a) + c
20.85 = 289a – 170a + c
20.85 = 119a + c
(5) – (4)
5.15 = 140a
a= = 0.0368 (6)
Substitute (6) into (1)
b= -10a = -10 x 0.0368 = 0.368
Substitute (6) into (5)
20.85 = 119(0.0368) + c
c= 16.47
Therefore, the quadratic equation is
y= 0.368x2 – 0.368x + 16.47
Ages (yrs) | BMI |
2 | 15.88 |
3 | 15.70 |
4 | 15.59 |
5 | 15.55 |
6 | 15.59 |
7 | 15.70 |
8 | 15.88 |
9 | 16.14 |
10 | 16.47 |
11 | 16.87 |
12 | 17.35 |
13 | 17.91 |
14 | 19.53 |
15 | 19.23 |
16 | 20.00 |
17 | 20.85 |
18 | 21.77 |
19 | 22.76 |
20 | 23.83 |
- On a new set of axis, draw your model function and the original graph. Comment on any differences. Refine your model if necessary.
From the equation y= 0.368x2 – 0.368x + 16.47, I put all the points in a table with the same x axis but using the new y values.
I then plot both the original graph and my model function on the same set of axis so I can compare the values.
Blue line: Original graph
Red Line: Model function
From the graph above, it seems that the model function is quite similar to the Original graph, but is not close enough. This is because the y value of the original function tends to decrease as the x moves towards 20. The y value of the modelled function however, tends to increase greater as the x value moves towards 20.
Conclusion
From the graph, it can be seen that my model does not fit this new data and therefore I would have to change my model.
I tried to remodel the quartic function again as it best fits this new data.
The dotted line shows the BMI of Chinese women and the continuous line shows the quartic polynomial y= -0.0004x4 + 0.0131x3-0.0635x2-0.4129x+16.846
Conclusion
From looking at all these functions modelled, the quartic polynomial best fits the graph of Chinese women’s BMI.
After completing the task, I found a flaw to the quartic polynomial as it is only reasonable up to the age of 20. After that, the graph immediately descends. This is highly unreasonable as a women’s BMI at the age of 25 is 15.4. However, due to the insufficient data provided for the BMI of Chinese women, I have tried every single function I can find and there is still no solution to the issue. This may be resolved by using more data.
However, the quartic graph, is the most accurate compared to the Gaussian, sine, cubic and quintric graph for the BMI of Chinese women between the age of 1 to 18 and therefore, I will not make any changes but keep this modelled function for this data.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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