Gold Medal Heights

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Gold Medal Heights

Introduction:

The high jump has and always will be a sport of the Olympics. Athletes of the high jump push themselves harder and harder to achieve their best, physically possible jump height in preparation for competing in the Olympic games. With the use of the best–fit graph, this report will examine the high jumps over numerous Olympic years as well as for height predictions in the near future.

Below is the table of jump heights from the Olympic games between 1932 and 1980:

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|

This graph demonstrates the relationship between the years (the x–axis) and the jump heights (the y–axis) of the gold medalists in the Olympic games between 1932 and 1980. If the actual value of the years were applied into the graph without any change, it would have been too spacious and complicated to present the data. Therefore, to make things simpler, I decided to remove each year’s first two digits.

e.g.) 1932 = 32

With the use of the formula y = mx + c, the parameters are m and c.

One constraint of this task is that two data points on the graph were removed from the regular pattern, meaning that two possible data points were eliminated that may have contributed with the data collection. This was due to World War I; two Olympics were cancelled (8 years) which negatively affected jump height improvement causing 1948’s gold medalist to have a lower jump height than 1936’s gold medalist. Another constraint is that there are no negative years (x > 0, y > 0).

The function that best models the behavior of the graph would be a linear function due to the fact that a best–fit line on the graphing program almost resembles a straight line. Therefore, as said earlier, I will be using the equation y = mx + c.

By manually drawing in a straight line that I believe best–fits the graph, I have found two easy points (the blue points on the graph) that allow me to find the slope (m) for the equation y = mx + c.

Slope = (y2 – y1)
(x2 – x1)

Slope = (230 – 200)
(75 – 42.5)

Slope = (30)
(32.5)

Slope = 0.92

y = 0.92x + c

230 = 0.92(75) + c

c = 230 – 0.92(75)

c = 161

∴ y = 0.92x + 161

The graph above shows the comparison between the line of best fit (orange) and my own model (blue). There is an intersection between the two lines but over all, the two are similar. As shown, the two lines cross over the y–axis at the difference of 10 just that they lean into the points at different angles. Obviously, the fact that my model was human–made as I saw fit, it is better to rely on the orange line. However, based off the dramatic 9 cm increase from the 1980 Olympics high jump gold medalist, I believe that my graph has a little more accuracy to it, but at the same time, humans have their physical limitations so a gradual increase (the orange best–fit line) may also be accurate. Therefore I will refine my graph by redrawing and estimating a line between the two.

Refined graph: (black line)

I then must remake the equation to input into the graphing software from the estimated black dots.

Slope = (y2 – y1)
(x2 – x1)

Slope = (190 – 170)
(30 – 5)

Slope = (20)
(25)

Slope = 0.8

y = 0.8x + c

190 = 0.8(30) + c
c = 190 – 0.8(30)

c = 166

∴ y = 0.8x + 166

Another similar function to my redefined linear function is the exponential function where:

y = e (ax + b)

In this case, y = e (0.0035x + 5.1585)

The exponential function (orange line) intersects the y–axis 7 units higher than my redefined line (black line) but as the two intersect, they basically lie on top of each other having almost equal values from (55, 210) onwards. To find the values of the jump...
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