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High Jump Gold Medals Portfolio Type 2 Math

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Introduction

IB SL, Type 2 Math Internal

Gold Medal Heights

The aim of this task is to consider the winning height for the men’s high jump in the Olympic Games and derive an equation that would show the trend of winning heights from 1932-1980.

The table below (Table 1) shows the height (in centimeters) of the winning jump achieved by gold medalists in the Men’s high jump in the Olympic Games from 1932-1980. The Olympic Games are held every four years except in 1940 and 1944 when the Olympic Games where cancelled due to WWII. This table was made using Microsoft Excel. The points given have then been plotted on a scatter plot (Figure 1)

Table 1:

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

Figure 1:

This graph was produced using Microsoft Excel using the data points from Table 1 (above).

The Olympic Games of 1940 and 1944 where canceled due to WWII. This means that the data recorded does not increase in even four year intervals. The canceling of the two Olympic Games appears to have affected the pattern of the data.  It can be assumed that due to the break in the games for the war, high jumpers involved in the event would have had limited chance to practice and lack

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Middle

y=0.76x+194 (All figures rounded to 3sf)

Table 3 shows the data excluding that gathered in 1932 and 1936 with the Year row showing years since 1948

Table 3:

Year

0

4

8

12

16

20

24

28

32

Height (cm)

198

204

212

216

218

224

223

225

236

Figure 4: Points from Table 3 plotted on a linear graph with line of best fit generated by Graphical Analysis

Gradient: Using points B (4,204) and E (16,218)

   m= y2-y1x2-x1 = 218-20416-4=1412=76

Linear Equation Shown on Graph:

y-218x-16=76                                 → Cross multiply

y-218=76(x-16)                → Simplify

y-218= 76x-16                  →Put into form y=mx+c

y=76x+202

y=1.16x+202

Figure 5: The linear function y=1.16x+202n (as shown in Figure 4) compared to the ‘line of best fit’ generated by graphical analysis

Linear Equation generated using technology: y=0.76x+194

Linear Equation: y=1.16x+202

I used technology to refine the equation y=1.16 x+202 by finding the line of best fit using Geogebra which generated the equation in the form

-49x+48y=9648

To get the equation in the form y=mx+c:

-49x+48y=9648                                                                        Rearrange equation into form y=mx+c

48y=9648+49x                                                                          Divide whole equation by 48

y=1.02x+201 (All figures rounded to 3sf)

In comparison the linear model does not fit the Table 3 data (as shown in Figure 4 and 5) as well as it fits the data shown in Table 2  (as shown in Figure 2 and 3). The line of best fit generated using technology passes through two points in Figure 3

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Conclusion

Figure 10: Logarithmic model y=28.75 log⁡(2.82*106x) shown against data from Table 4 produced using Geogebra

image04.png

The graph above (Figure 10) shows the points given (Table 4) modeled by the logarithmic equation: y=28.75 log⁡(2.82*106x)Geogebra was used to produce the graph.

Figure 10: Gaussian function shown against data from Table 4 produced using Geogebra

image05.png

Because of my limited knowledge of Gaussian functions I used technology to produce the graph. I wanted a function that would show the decrease in the slope reflecting the lessening rate of improvement. The Gaussian function has the steady slope similar to that of the linear function that accurately reflects the trend of the data over the middle of the 20th century and has a decreasing gradient similar to that of a logarithmic function near the end of the 20th century as humans approach the limit imposed on them by natural forces such as gravity. This function is limited in its use however as it would show a gradual decrease in winning high jump heights if projected into the future and there is no current evidence to suggest that this would be the case. This graph has the equation:

y=7.503*105 -1152x+0.5895x2  -1.005*10-4x3

In conclusion the equation y=7.503*105 -1152x+0.5895x2  -1.005*10-4x3  seems best to illustrate the trend of winning high jump heights of men in the Olympic Games between 1932-1980 while fitting the data both before and after this period with accuracy.

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