• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11

# Gold Medal Heights Maths Portfolio.

Extracts from this document...

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

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.

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.

Since we know that the d-value is 250,

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:

=

Point 2:

=

Point3:

=

Subtract equation 2 from 1

Substitute

into equation 3

Substitute equation 4 and

into equation 1

46.469

Substitute

46.469 into equation 4

Figure 3: Exponential and Cubic Models

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.

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/>.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## This portfolio is an investigation into how the median Body Mass Index of a ...

Under this assumption, the value found is very reasonable. Also under this assumption, one can conclude based on this data that the median BMI of a woman who has reached both adult height and adult weight would be a value very close to the asymptote of the function - 22.9.

2. ## Extended Essay- Math

ï¿½ ï¿½}ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½hï¿½ï¿½ï¿½)))ï¿½ï¿½-/O2r K|!u(tm)ï¿½ %>ï¿½ï¿½/Vj+Vï¿½ï¿½("VAdtï¿½ï¿½Iï¿½-ï¿½Yï¿½Ö­;{ï¿½ï¿½Eï¿½Ο?ï¿½""dï¿½ï¿½k ï¿½ï¿½Pn~ï¿½B^D"ï¿½9&ï¿½Hï¿½yKï¿½ï¿½(c))(r) Yï¿½ "#ï¿½Vï¿½Zï¿½Ëï¿½&Mï¿½6=|ï¿½ï¿½9lï¿½öm^ï¿½ï¿½ï¿½Mï¿½G(Zï¿½5Pï¿½Bï¿½0!ï¿½ï¿½ï¿½Kï¿½ï¿½0Lï¿½"#Xï¿½ï¿½;ï¿½gï¿½-tï¿½mï¿½.]ï¿½/^?ï¿½fï¿½9&J-Zï¿½ >l! Kï¿½lï¿½ Ù1/2#' xï¿½ ï¿½4ï¿½ï¿½Nï¿½<ï¿½Ó¬Y3ï¿½R(tm)ï¿½3vï¿½ï¿½(r)]"ï¿½ï¿½ï¿½...pMï¿½ï¿½ï¿½ÍCï¿½&ï¿½>.ï¿½~Û¶mï¿½xï¿½ ]|ï¿½ï¿½bï¿½ï¿½...-Uï¿½>-ï¿½(ï¿½+lï¿½Z\$ï¿½a1/2Yï¿½ Ð§Oï¿½^1/2zaÉ£#8ï¿½ä·¿ï¿½-Ìï¿½ï¿½ï¿½Q~ï¿½ï¿½ï¿½-iï¿½ï¿½Eï¿½ e"ï¿½\$y1Z?ï¿½ ï¿½k]Bd#aï¿½ï¿½daF'Hdï¿½\""ï¿½ï¿½ ï¿½"cG ï¿½mï¿½ï¿½ëï¿½#ï¿½ CKxpHFZ3/4ï¿½Zï¿½qï¿½Qdï¿½ï¿½Zï¿½Oï¿½1/2Aï¿½dï¿½ (r)-ï¿½ï¿½]ï¿½l\$-ï¿½(ï¿½ 8(Dï¿½bI0 i(c)ï¿½}ï¿½ï¿½Uï¿½PAKï¿½C"ï¿½Jï¿½ G!xï¿½'dï¿½ï¿½ï¿½eï¿½ï¿½#ï¿½[nï¿½eï¿½ï¿½ï¿½Z*ï¿½B{ï¿½ï¿½ï¿½{ï¿½ï¿½Úï¿½ï¿½+(r)Zï¿½ (r)ï¿½ï¿½*hQ ï¿½Wï¿½nï¿½3/4][email protected]ï¿½HHÎ4Rï¿½EQï¿½ï¿½ï¿½,d\ï¿½3/4Cï¿½ ï¿½ï¿½ï¿½ï¿½Fb''7ï¿½zï¿½*)ï¿½|ï¿½^v"ï¿½mï¿½(c)qï¿½&kï¿½"!)ï¿½ï¿½'9ï¿½ ï¿½"sX8ï¿½zï¿½ï¿½Õ¡(tm)ï¿½#ï¿½Ë @2ï¿½Oï¿½ï¿½ï¿½ï¿½iiiï¿½ï¿½Mb\$rï¿½ï¿½+ï¿½hï¿½(r)ï¿½dï¿½ï¿½ï¿½á£"ï¿½:tï¿½ï¿½ï¿½R \$ï¿½ï¿½\$#ï¿½Tï¿½}} wdï¿½ï¿½ï¿½ï¿½B#ï¿½\$#Í´ï¿½ï¿½p-ï¿½(tm)ï¿½'""4"Ùè¸\$#@ï¿½ï¿½Iï¿½Oï¿½z]ï¿½D /;ï¿½6Vj ï¿½ ï¿½zï¿½'ï¿½ï¿½ï¿½I\ï¿½0ï¿½ï¿½7sHï¿½'ï¿½;2ï¿½*'iï¿½n\$xlÜ¸ï¿½fBï¿½&.Ò¢ï¿½ï¿½wf ï¿½ï¿½ï¿½ï¿½3/4o iï¿½Z8ï¿½/xï¿½tï¿½ï¿½JMï¿½wï¿½mqIFn#ï¿½q{ï¿½ï¿½ D6\$ï¿½ï¿½Ö¬YYï¿½'ï¿½ ï¿½@Yï¿½t Oï¿½2eï¿½dff"g"Nr' +rWÖ¯_ï¿½ï¿½1/2Zgï¿½K` ï¿½ÎbBï¿½ï¿½ lï¿½c+ï¿½Lï¿½ï¿½"_ï¿½Jnï¿½+ï¿½ï¿½ ï¿½d\$ j:ï¿½ï¿½ï¿½||1/4]ï¿½]xEtnï¿½dï¿½"ï¿½ï¿½ï¿½ï¿½ï¿½Hï¿½qï¿½]ï¿½#a N ï¿½H']gddT"VM'ï¿½ï¿½È(tm)ï¿½ï¿½ï¿½ï¿½*ï¿½(tm)ï¿½ï¿½'ï¿½tR'L*ï¿½dï¿½"ï¿½vd...ï¿½yï¿½ï¿½ï¿½--ï¿½ï¿½ï¿½ ''Nï¿½Ä¬ï¿½ï¿½×¡ ï¿½Ñ£ï¿½ |Zï¿½ï¿½ï¿½ @2ï¿½Fï¿½0!ï¿½ï¿½/ï¿½ï¿½,`ï¿½ï¿½ï¿½ï¿½ï¿½^ ''6ï¿½ï¿½ï¿½ï¿½Tï¿½9Aï¿½ï¿½ï¿½ï¿½V(ï¿½{ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½[email protected]ï¿½Cï¿½ï¿½ï¿½ï¿½#N

1. ## A logistic model

1.6u ?5 2 un?1 ? (?1? 10 )(un ) ? 1.6un {3} 3. Using the model, determine the fish population over the next 20 years and show these values using a line graph One can determine the population of the first 20 years just by knowing that u1=1?104 u2=1.5?104 .

2. ## Maths IA Type 2 Modelling a Functional Building. The independent variable in ...

: 1 The efficiency is seen to be the same as when the fa�ade was on the shorter side of the base. After determining the efficiency, we find the area of the cuboid floor once again. Table 7. Area of cuboid floor when the fa�ade switches sides Maximum height of roof (m)

1. ## High Jump Gold Medal 2012 maths investigation.

After sketching the new model function using the original data using Geogebra. The gradient calculated by the software is represented by : y= 0.7456x+194.2535 Testing the software linear equation: y= 0.7456x+194.2535 Let x = 44, we expect y= 225. y= 0.7456 (44)+ 194.2535 y= 227.0599 There is a bit different from the actual data.

2. ## Mathematics Portfolio. In this portfolio project, the task at hand is to investigate the ...

That there is a scaling factor that moves the value of time up as it increases. There is already a trend when looking at how t increases. Each consecutive value seems to rise by .02 and .07, for which the values then seem to move a decimal place to the

1. ## Population trends. The aim of this investigation is to find out more about different ...

The sine curve starts to assimilate the points but it requires to be shifted towards the data points. The current equation is , from the previous graph I learnt that the curve had to be shifted upwards and to do so the had to be increased.

2. ## IB Math Methods SL: Internal Assessment on Gold Medal Heights

197 203 198 204 212 216 218 224 223 225 236 Now we can begin with our Least Squares approach. The Least Squares approach involves finding the best-fit equation using sums and arithmetic of the data. It enables us to analytically to find an equation that models the data of our table above.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work