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Gold Medal Heights Maths Portfolio.

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Introduction

Gold Medal Heights

The table below gives the data achieved by the gold medalists in the Olympic Games.

Year

1932

1936

1948

1952

1956

1960

1964

1968

1972

1976

1980

Height(cm)

197

201

198

204

212

216

219

224

223

225

234

Using this data, I will create a model to represent the relationship between the high jump results and the years they took place. However, the Olympic Games did not occur in 1940 and 1948 due to World War 2. The independent variable is time, so let x-years be time and the dependent variable is height, so let y-centimetres be height. The winning height is the dependent variable since each year resources, technology, competition and may more are altered. After plotting the data, the following scatterplot was obtained:

Figure1: Height vs. Year        Figure1.1: Window

image10.pngimage11.png

Since x is in-terms of years, one may consider B.C as “negative” years.

 the domain is

. The range is in-terms of the height jumped by the medalist therefore, theoretically speaking it is impossible for humans to jump above a certain height because we have our limitations and we cannot jump below a height of 0.

 the range is

. The maximum height that can be jumped is set at 250cm because the highest achieved height by a human is 245cm by Javier Sotomayor.

...read more.

Middle

 Another function that could model these points is an exponential.

Figure 2: Exponential Function: Applying Reflections and Translations

When an exponential is reflected on the x and y-axis, the shape is similar to the plotted points. It is then translated d-units up, where d would represent the horizontal asymptote. Since e, Euler’s number, is a transcendental constant, I defined a as e and added –k to create a similar graph.

image03.png

Since we know that the d-value is 250,

image03.png

image03.png

Since this function has three variables, three points must be chosen. To determine the first point, the first four data points are averaged, then the next four points are averaged to find the second point and finally, the last three data points are averaged to find the third point.

Point 1: image01.png

=

Point 2:image01.png

=

Point3: image01.png

=

image03.png

image03.png

image03.png

image06.png

image02.png

Subtract equation 2 from 1

image02.png

image07.png

Substitute

 into equation 3

image08.png

image00.png

image00.png

image03.png

Substitute equation 4 and

 into equation 1

image00.png

image03.png

image02.png

46.469

Substitute

46.469 into equation 4

image03.png

image03.png

Figure 3: Exponential and Cubic Modelsimage12.png

Both equations have their own limitations. The cubic function is not the best model because as

,

 and as

,

 the cubic function does not fit the range, but it fits the given data points well since the RMSE value for the cubic is 4.614.

...read more.

Conclusion

Works Cited

"High Jump World Records." Rob's Home of Sports, Fitness, Nutrition and Science. N.p., n.d. Web. 23 Feb. 2012. <http://www.topendsports.com/sport/athletics/record-high-jump.htm>.

"Logistic Equation -- from Wolfram MathWorld." Wolfram MathWorld: The Web's Most Extensive Mathematics Resource. N.p., n.d. Web. 22 Feb. 2012. <http://mathworld.wolfram.com/LogisticEquation.html>.

"Logistic function - Wikipedia, the free encyclopedia." Wikipedia, the free encyclopedia. N.p., n.d. Web. 22 Feb. 2012. <http://en.wikipedia.org/wiki/Logistic_function>.

"Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)." Eumetcal. N.p., n.d. Web. 23 Feb. 2012. <http://www.eumetcal.org/resources/ukmeteocal/verification/www/english/msg/ver_cont_var/uos3/uos3_ko1.htm>.

"NOAA/ESRL Global Monitoring Division - THE NOAA ANNUAL GREENHOUSE GAS INDEX (AGGI)." NOAA Earth System Research Laboratory. N.p., n.d. Web. 24 Feb. 2012. <http://www.esrl.noaa.gov/gmd/aggi/>.

...read more.

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