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High Jump Gold Medal 2012 maths investigation.

Extracts from this document...

Introduction

Gold Medal Heights

In this task I will develop a function that best fit the data points in the graph, which will be plotted based on the table below showing the different gold medal heights.

Year	1932	1936	1948	1952	1956	1960	1964	1968	1972	1976	1980
Height (cm)	197	203	198	204	212	216	218	224	223	225	236

Number of years start from 1932	Height (cm)
X-axis	Y-axis
0	197
4	203
16	198
20	204
24	212
28	216
32	218
36	224
40	223
44	225
50	236

Note: There are no data of 1940 and 1944

For all the graphs in this table, the y- axis will be represent the height in cm and the x-axis will be the year when the height was obtained.

The graph below shows the relationship between the years and the heights obtained between the years of 1932 and 1980:

The further explanation for the missing data in 1940 and 1944 was due to World War II. Although the data does not tell us the reason why the height increased in 1932 and 1936 and drop down abruptly in 1948, we can assume that the World War II had affected athletics health critically.

Middle

1932-1940 = 8 years

y= 217.9381 + 19.2462 sin (0.054x- 1.758)

x= 8

y =199.27 cm

1932-1944= 12 years

y= 217.9381 + 19.2462 sin (0.054x- 1.758)

x= 12

=> y = 200.7 cm

The answer would be mathematical reasonable increasing if we assume the data follows sine equation. The data are still in human range that the athletic can perform, however it is hard to say that they are 100% correct since we only rely on technology.

I also decided to use the function to predict the result in 1984 and 2016

1932 – 1984 = 52 years

y= 217.9381 + 19.2462 sin (0.054x- 1.758)

y= 234.63 cm

1932- 2016 = 84 years

y= 217.9381 + 19.2462 sin (0.054x- 1.758)

y=224.8 cm

The answer are mathematical correct. However, in reality we do not know whether the data will truly describe the result since we only the data up to 1980.

Data table from 1896 – 2008:

Numbers of year	Actual height/cm
0	190
8	180
12	191
16	193
24	193
32	194
36	197
40	203
52	198
56	204
60	212
64	216
68	218
72	224
76	223
80	225
84	236
88	235
92	238
96	234
100	239
104	235
108	236
112	236

The graph below represents the data from 1896 to 2008 :

Conclusion

an class="c11">227.3

197.6

190.2

216.0

223.1

237.3

194.1

223.1

192.5

194.1

220.7

192.5

237.6

220.7

218.6

237.6

Different between the actual data and data calculated by using

y= 212.9337 + 24.6777 sin (0.0324x- 1.9601)

y= 212.9337 + 24.6777 sin (0.0324x- 1.9601)	Actual height	Difference
190.1	190	0.1
227.5	180	47.5
236.1	191	45.1
210.9	193	17.9
201.8	193	8.8
233.9	194	39.9
206.1	197	9.1
188.3	203	-14.7
231.0	198	33.0
201.6	204	-2.4
188.8	212	-23.2
211.1	216	-4.9
236.1	218	18.1
227.3	224	3.3
197.6	223	-25.4
190.2	225	-34.8
216.0	236	-20.0
237.3	235	2.3
223.1	238	-14.9
194.1	234	-39.9
192.5	239	-46.5
220.7	235	-14.3
237.6	236	1.6
218.6	236	-17.4

After comparing the data between the actual provided height and the new best-fit model when we have addition data, it seems to be likely describing the nature of the data. However, the function does not fit in most of the data. Moreover, a better function should be investigated to make sure it goes through all the data point. However, due to the limit knowledge at the moment when I only know about linear, exponential, sine, logistic functions, it prevents me from going further to develop a more advance function which fit all the data points. Although the two functions have their R-values near to 1, especially sine function, it does not go through all the data point.

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

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