Sue Yoon IB Math SL: Period 3 Mrs. BessetteThis report will investigate the best-fit graph for the winning height for the men’s high jump inthe Olympic Games. Also, create a new equation to predict the winning height in the future.The table blow gives the height (in centimeters) achieved by the gold medalists at variousOlympic Games. Year 32 36 48 52 56 60 64 68 72 76 80 Height 1.97 2.03 1.98 2.04 2.12 2.16 2.18 2.24 2.23 2.25 2.36 (cm)Note: The Olympic Games were not held in 1940 and 1944.Using technology, plot the data points on a graph. Define all variables and used and state anyparameters clearly. Discuss any possible constraints of the task.
Sue Yoon IB Math SL: Period 3 Mrs. BessetteSince the numbers are big, to calculate easily, I ignored first two digits which are‘19’ and justuse last two digits. Ex) 1932 => 32 1936 => 36Also, I scaled height so that I can calculate more easily. Ex) 203= 100(2.03) y = 2.03Parameters:In this equation, a represents a horizontal shift b represents a vertical stretch.Possible constraints of this equation are thatWhat type of function models the behavior of the graph? Explain why you chose this function.Analytically create an equation to model the data in the above table.The height mainly increases as years pass. Since it doesn’t increase rapidly so I think the best fitfor this graph is the natural logarithm rather than using exponential.The natural logarithm equation is,To make my own equation, I used matrix. I picked two numbers which are P1 (32, 1.97) and P2(80, 2.36) because P1 is the first year and P2 is the last year. So I thought it can represent thewhole year.
Sue Yoon IB Math SL: Period 3 Mrs. BessetteThen calculate ln(x) part, 1 3.4657 a 1.97 = 1 4.3820 b 2.36 a 1 3.4657 197 = b 1 4.3820 2.36 a 0.4949 = b 0.4256
Sue Yoon IB Math SL: Period 3 Mrs. BessetteOn a new set of axes, draw your model function and the original graph. Comment on anydifferences. Discuss the limitations of your model. Refine your model if necessary.Some points such as x=48 or x= 52 are quite off from the line. So I picked different number. Iused Median-Median method. Since the number of data is odd number, I decided to ignore themiddle number which is ‘60’. Year 32 36 48 52 56 60 64 68 72 76 80 Height 1.97 2.03 1.98 2.04 2.12 2.16 2.18 2.24 2.23 2.25 2.36 (cm) 32 48 36 52 56 64 72 68 76 80 1.97 1.98 2.03 2.04 2.12 2.18 2.23 2.24 2.25 2.36 Median
Sue Yoon IB Math SL: Period 3 Mrs. BessetteAfter I get the two points which are P1 (36, 2.03) and P2 (68, 2.24) again, use matrix to make myown equation.Then calculate ln(x) part, 1 3.5835 a 2.03 = 1 4.2195 b 2.24 a 1 3.5835 2.03 = b 1 4.2195 2.24 a 0.8468 = b 0.3302
Sue Yoon IB Math SL: Period 3 Mrs. BessetteAfter I changed the number, the RMSE has decreased by 0.0248.Use technology to find another function that models the data. On a new set of axes, draw bothyour model functions. Comment any differences.I used linear equation to compare with natural logarithm equation. Also, I checked RMSE (RootMean Square Error). If the value of RMSE is lower, it means the graph is more accurate.The graph above shows linear function.RMSE: 0.04523
Sue Yoon IB Math SL: Period 3 Mrs. BessetteThe graph above shows the auto fit of natural log function.RMSE: 0.05681The graph above shows the manual fit of natural log function.RMSE: 0.05681Even though RMSE of linear function is lower than RMSE of natural log function, since thelinear function increases rapidly, it is better to use natural log function.
Sue Yoon IB Math SL: Period 3 Mrs. BessetteThe graph above shows auto fit of natural log function and manual fit of natural log function.The RMSE of auto fit is 0.05681 and the RMSE of manual fit is 0.0716. Even though there is a slightdifference between manual fit and auto fit, it seems like two equations are almost same sinceRMSE is similar.Had the Games been held in 1940 and 1944, estimate what the winning heights would have beenand justify your answers.In 1940:When I use my own equation, the answer would be,
Sue Yoon IB Math SL: Period 3 Mrs. BessetteIn 1944:When I use my own equation, the answer would be,Use your model to predict the winning height in 1984 and in 2016. Comment on your answers.In 1984:When I use my won equation, the answer would be,
Sue Yoon IB Math SL: Period 3 Mrs. BessetteIn 2016:When I use my won equation, the answer would be,As year goes, the winning height for the men’s high jump in the Olympic Games keepsincreasing. Thus, it shows that this formula can’t be used after certain year.The following table gives you the winning heights for all the other Olympic Games since 1986. Year 1896 1904 1908 1912 1920 1928Height (cm) 190 180 191 193 194 235 Year 1984 1988 1992 1996 2000 2004 2008Height (cm) 235 238 234 239 235 236 236Again, since the numbers are big, I ignored first two digits which are‘18’, ‘19’ and ‘20’ and justuse last two digits to calculate it easily. I set the numbers based on ‘19’, ‘18’ should be negativenumber and ‘20’ should be more than 100. Ex) 1896 => -4 1936 => 36 2004 => 104Also, I scaled height so that I can calculate more easily.
Sue Yoon IB Math SL: Period 3 Mrs. BessetteHow well does your model fit the additional data?As the graph above shows, natural logarithm equation doesn’t really fit well as years pass. Thedots tend to be off from the line. Also as year passes, the graph will keep increasing which isimpossible for human.Discuss the overall trend from 1896 to 2008, with specific references to significant fluctuations.Overall, from 1896 to 2008, the height achieved by the gold medalists increases. However, atsome points, it decreases. For example, 1896 to 1904 it decreases by 10cm and after 1904, itstarts to increase again. However, in 1992 and 2000, it dropped.
Sue Yoon IB Math SL: Period 3 Mrs. BessetteWhat motivations, if any, need to be made to your model to fit the new data?Since the natural log equation doesn’t really fit well, I used the cubic equation for the additionaldata and it seems to be fit well.Academic Honesty“I, the underdesigned, hereby declare that the following assignment is all my own work and thatI worked independently on it.”“In this assignment, I used Logger Pro to draw my graphs.”