Gold Medal Heights Maths Portfolio.

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Gold Medal Heights

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

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

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.

As the graph suggests that there is a positive relationship between the height and the year. Functions which could model this situation where it increases at a decreasing rate include reciprocal, logarithm, exponential to the power of a number between 1-0 etc.

 A reasonable polynomial function which can represent the data is a cubic  

Since a cubic has four variables, four points are necessary to solve for the co-efficient. To determine the first three points, three consecutive data points were averaged starting from 1932 to 1972, the last point was determined by the average of the last 2 points.

Point 1:

=

Point 2:

=

Point 3:

=

Point4:

=

1.

2.

3.

4.

Subtract equation 1 from 2

5.

Subtract equation 3 from 4

6.

Subtract equation 5 multiplied by 10 and equation 6 multiplied by 18

Join now!

7.

Substitute equation 7 into 1

8.

Substitute equation 7 into 2

9.

Subtract equation 8 multiplied by

 and equation 9 multiplied by

10.

Substitute equation 10 and 7 into equation 3

11.

Substitute equation 10, 7 and 11 into equation 4

Substitute equation 11

Substitute

 into equation 7

Substitute

 into equation 11

...

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