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# High Jump Gold Medal 2012 maths investigation.

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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

76

197.6

80

190.2

84

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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