Math Gold Medal

7/31/2019 Math Gold Medal

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Introduction

a) The Olympic

Games is aninternational

event featuringsummer and

winter sports,in which

athletesparticip


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ate in different

competitions.

In ancientGreece the

Olympic Gameswere athletic

competitionsheld in honor of

Zeus. Since the


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Olympic Games

began they have

been thecompetition

grounds for

greatestathletes. First

place obtaining


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gold; second

silver and third

bronze. TheOlympic

medalsrepresent the hardship

of what thecompetitors of

the Olympics


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have done in

order to obtain

the medal.Onone side the

Olympic medalhas Nike the

goddess ofvictory holding

a palm and a


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winners crown

andon the other

side the medalhas a different

label for eachOlympiad

reflecting thehost of the

games.Olympic


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medals could be

used as a unit

of measure ofathleticism.Top

10 OlympicMedal- winning

CountriesCountry Medals won1.


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The

UnitedStates24

042.

Soviet Union

12043.

Great Britain

6894.

France 6795.Germany 6486.


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Italy 5957.Sweden 5888.East Germany5199.Hungary 45410.

Finland 446This

is a tableshowing the top

10 Olympic


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Medal- Winning

Countries and

definitelyshows how

theOlympicscan be seen as a

standardizedunit of

athleticism


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Dataa)Height (in

centimeters)

achieved by the

gold medalist at

various Olympicgames.Year

1932 1936 1948


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1952 1956 1960

1964 1968 1972

19761980Height(cm)

197 203 198204 212 216

218 224 223225 236


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Variables andConstraintsa) Thedependent

variable forthis data set is

the OlympicGold Medalist


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Heights. The

independentvari

able for thisdata set is the

years in whichthe summer

Olympic Gamesoccurred in. A

constraint of


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this data set is

the limited

amount of datathat is

available. Thedata available is

only between1932 and


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19520020521021

5220225230235

2401 9 2

0 1 9 3 0

1 9 4 0 1

9 5 0 1 96 0 1 9 7

0 1 9 8 01 9 9 0 H


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e i g h t ( c m )YearYear Of

OlympicsVS. Gold

Medal


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H

eightsInitial values1980. If there

were more data

and if there is a

pattern the

pattern would


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become more

apparent. Ontop

of this there isa gap between

1936 and 1948which is a 12

year chunk ofdata missing.

And since


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theOlympics

are held every

4 years that is3 Olympic

competitionsmissing from

the data.b) In amath textbook

the variables


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and constraints

could be seen

as y= OlympicGold Medalist

Heightsand x=years in which

the summerOlympic games

occurred in.c)


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In the context

of this problem

is that the xaxis would be

used to showthe Year of the

OlympicGamesand y

would be used


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to show the

height of the

gold medalist.d)This data set is

continuousbecause it is

associated witha measurement

and its possible


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to have

thesame y value

for different xvalues. And

since the datais measuring

height adecimal answer

is possible.A


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function that

would fit most

of the datawould be a

quadraticfunction. The

constraint thatthere is a12

year gap


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between 1936

and 1948 could

skew the data.The data that

we are missingcould of

showedus amuch clearer

model like a


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linear or could

of reassured us

of a quadraticmodel.Graph ofInitialD

ataa)


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Analysis andModel

ConstructionBased on thedata the typeof curve that

might beexpected is


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quadratic and

maybe even a

thirddegreefunction.

The expectedshape would be

quadratic if wedisregarded the

1936 value and


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itlooks like a

third degreefunction would

fit well.

a)Some generalformulas that

the data could


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fit can be

quadratic


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f(x),linear


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, or exponential


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.Quadratic

possible values


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:X Y1932 197

begin_of_the_s

kype_highlighting 1932 197

end_of_the_skype_highlightin

g.991936199.041948

204.331952


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206.821956

209.671960

212.881964216.461968

220.391972224.691976

229.341980234.36Linear

Possible values


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X Y1932 194

begin_of_the_s



g.141936197.161948

206.221952


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209.241956

212.261960

215.281964218.31968

221.321972224.341976

227.361980230.38Exponen


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tial Possible

Values


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X Y1932 194

begin_of_the_skype_highlighti

ng 1932 194end_of_the_sk


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ype_highlightin

g.721936

197.491948206.041952

208.971956211.951960 214begin_of_the_s

kype_highlighti


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ng 1960 214

end_of_the_sk

ype_highlighting.961964

218.021968221.121972

224.271976227.461980

230.69All of


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the formulas

used to fit the

data areincreasing and

all have a min (when x=0) for

thequadraticfit the minimum

is 41946.847 of


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the linear fit

the minimum is

-1264.6484 andfor

theexponentialfit the min is

0.21211804. Allof the formulas


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maximum.For

the model to

become muchmore realistic

the logical fitwould be a

sinusoidalcurve.

Theconstraints


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on a sinusoidal

curve would be

that the curvewould have to

be half a cycle.The curve has

tobe half acycle because it


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curve would

show that the

heightsfluctuate from

OlympicstoOlympics and

this would notrepresent our

data. But by


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limiting the

curve to half a

cycle then thecurvewould fit

the data.b)

Linear Model

QuadraticModel


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General formfor a linear

equation:


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When using twopoints from thedata

andsubstituting

them into the


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values for x

andy the values

for m and b canbe found.

Thepoints thatare going to be

used aregoingto be

(1972,223) and


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(1960,216).

Thesepoints

were chosenbecause they

look likea niceline can be

drawn betweenthemwithout

too much


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differentiation

betweenthe

other points.


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To find the

value of a, we

can subtractthetwo

equations. Thiswill


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We can now

find m by

dividing bothsidesby 12 so

we get:


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The value for b

can now be

foundbysubstituting


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to any of theequations


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The general

form for an

quadraticequation is


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. With 3pointson the graph anequation can

beformulated in

the form


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. The points

that are going

to be usedaregoing to be

(1936,203),(1972,233)

and(1960,216)We can make

three equations


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using

thesethree

points and thegeneral formula


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. bysubstitutingxand y values

we get:


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or


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or


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or


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Then we cansubtract twoequationstogeth

er to eliminate

C. I chose to


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subtractthe

first and

second equation


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The equationwill now be:


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Now I willsubtract thesecond and

thirdequation

to get two new


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equations

tosolve for a

and b.


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With the two

equations now

we mustisolatea variable to

solve, I will bechosinga, so we

need toeliminate b. we


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can dothis by

multiplying


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Bysince


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. By multiplyinby

we get


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. then weadd

the equations:


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Now we can

solve for a and

we get


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. Now we can

substitute ainto

one of theequations with

twovariables tosolve for


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. I will be using

theequation


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.bysubstitutinga we get:


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we can nowsubstitute botha and b intothe


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original

equations and

solve for c. Iwillbe

substituting ininto the

secondequation.


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Now we havefound allvariables and


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wecan write our

equation. And

we get:


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Linear FunctionQuadraticfunction


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The reason whyI chose a linearmodel is


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because despite

some points the

data in thegraph seemed

tobe modeledwell by a linear

model. I alsochose a

quadratic model


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because the

data points

seemed tobemodeled well

by a quadraticmodel.b)Linear Equation:


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1952002052102152202252302352401 9 2

0 1 9 3 01 9 4 0 1


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9 5 0 1 9

6 0 1 9 7

0 1 9 8 0

1 9 9 0 He i g h t ( c m )

Year


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Year Of

Olympics

VS. GoldMedalH


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eightsInitialvaluesLinearEquationYearsInitialvalues Linear

Equation1932197 199.671936


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203

202.003333319

48 198209.003333319

52 204211.336666719

56 212213.671960 216

216.003333319


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64 218

218.336666719

68 224220.671972

223223.003333319

76 225225.336666719

80 236


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227.67This

linear function

is not the bestfit for the

model becauseas we can see

from the graphthere are a

lotof points left


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out and that

are not even

close to thelinear equation.

Also in the datatable there

aresome yearswhere the

linear equation


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varies greatly

from the actual

seethat forsome years it is

close like for1936 but for

others it is


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much farther

away from the

linearequationlike in 1948.0501001502002503001 9

2 0 1 9

3 0 1 9

4 0 1 9


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5 0 1 9

6 0 1 9

7 0 1 9

8 0 1 9

9 0 He i g h t ( c m )Year


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Year Of

Olympics

VS. GoldMedalH


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eightsQuadraticEquation:


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Years Initialvalues

Quadratic

Equation1932

197

204.578947419

36 203


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2031948 198

204.157894719

52 204206.508771919

56 212209.842105319

60 216214.157894719

64 218


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219.456140419

68 224

225.73684211972 223

2331976 225241.245614198

0 236250.4736842T

he quadratic


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equation that I

came up with is

a reasonable fitbut still not the

best fit. As wecan seefrom

the graph thequadratic

equation


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increases

quicker than

the actualvalues. This

causes the datato0501001502002501 9 2 0

1 9 3 0 1


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9 4 0 1 9

5 0 1 9 6

0 1 9 7 0

1 9 8 0 1

9 9 0 He i g h t ( c m )Year


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Year Of

Olympics

VS. GoldMedalH


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eightsInitialvaluesLinearRegressionbe way off forthe final two

years in ourdata. We can


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see this from

the data table

as 1976 and1980 arenot as

close are theother actual

values to thevalues from the

quadratic


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equation.Linear

Regression


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Years Initial

values Linear

Regression1932197

194.13824261936 203

197.15850481948 198

206.219291419


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52 204

209.239553619

56 212212.259815819

60 216215.280078196

4 218218.3003402


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19520020521021

5220225230235

2401 9 2 0

1 9 3 0 1

9 4 0 1 9

5 0 1 9 60 1 9 7 0

1 9 8 0 19 9 0 H


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OlympicsVS. Gold

Medal


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H

eightsInitial

valuesQuadratic

Regression1968 224

221.32060241972 223


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224.340864619

76 225

227.36112681980 236

230.381389This is a better

linear model butit is still not

the best.


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Because it is

linear it

excludes somepoints like1948.

In 1948 we cansee that

differencebetween the

actual values


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and the linear

regression

models isquitesignificant.Qua

draticRegression


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Years Initial

values

QuadraticRegression1932

197197.995296519

36 203199.0379511194

8 198


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204.334820919

52 204

206.82341271956 212

209.67348881960 216

212.88504931964 218

216.458094


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19520020521021

5220225230235

2401 9 2 0

1 9 3 0 1

9 4 0 1 9

5 0 1 9 60 1 9 7 0

1 9 8 0 19 9 0 H


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OlympicsVS. Gold

Medal


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H

eightsInitial

valuesQuadratic

Regression1968 224

220.3926231972 223


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224.688636419

76 225

229.3461341980 236

234.3651159This is the best

fit because aswe can see

from the graph


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it fits most

points without

leaving othertoo faroff.

From the datatable we can

see that this isthe only

function that


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does not vary

too wildly from

theactual value.

Proposed

modelThe quadraticregressionmodel is a


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reasonable fit

for the data

because theregression line

does not varyallthat much from

the actualvalues.As we

can see the


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quadratic

regression

model fits thegraph almost

perfectly. Theonly year that

thegraph doesnot fit all that

well is 1948.


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The data is

increasing for

the entiredomain but this

doesnotnecessarily

mean thatfuture years

will be


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accurate. This

is because

there arealmost

asymptotes inall of the

graphs. Therehas to be a limit

of how low the


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heights have to

be to qualify

and since theOlympiccompeti

tors are humanthere is a limit.

This makes allof the graphs

not a best fit


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for all of the

futureyears

that the eventwill be held and

all of the pastyears but this

quadratic modeldoes model the


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datagiven

accurately.Considerationof AccuracyLinear Equation:


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Years Initial

values Linear

EquationPercent

Error1932 197199.67

1.351936 203202.0033333

.491948 198


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209.0033333

5.551952 204

211.33666673.61956 212

213.67 .791960216

216.0033333.00151964 218

218.3366667


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.1541968 224

220.67

1.481972 223223.0033333

.00151976 225225.3366667

.151980 236227.67

3.53Average


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1.55Quadratic

Equation:


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X-Values Initialvalues

Quadratic

Equation Error

Percent1932

197 204

begin_of_the_s


198/306

kype_highlighti

ng 1932 197

204end_of_the_sk

ype_highlighting.5789474

3.851936 203203 01948 198

204.1578947


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3.111952 204

206.5087719

1.231956 212209.8421053

1.021960 216214.1578947

.851964 218219.4561404

.671968 224


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225.7368421

.771972 223

233 01976 225241.245614

7.221980 236250.4736842

6.13Average2.26Linear

Regression:


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X-Values Initialvalues LinearRegression


203/306

Error

Percent1932

197 194begin_of_the_s


194end_of_the_sk


204/306

ype_highlightin

g.1382426 1.451936 203 197begin_of_the_s


197end_of_the_sk

ype_highlightin


205/306

g.1585048

2.881948 198

206.21929144.151952 204

209.23955362.561956 212

212.2598158.1221960 216

215.280078


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.3331964 218

218.3003402

.141968 224221.3206024

1.21972 223224.3408646

.601976 225227.3611268

1.051980 236


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230.381389

2.38Averga

1.53QuadraticRegression:


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X-Values InitialvaluesQuadratic

Regression

Error


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Percent1932

197 197

begin_of_the_skype_highlighti

ng 1932 197197


g.9952965


212/306

.501936 203

199.0379511

1.951948 198204.3348209

3.21952 204206.8234127

1.381956 212209.6734888

1.11960 216


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212.8850493

1.441964 218

216.458094.711968 224

220.3926231.611972 223

224.6886364.761976 225

229.346134


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1.931980 236

234.3651159

.69Average1.36From the

errorpercentages we

can see thatthe most

accurate is the


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quadratic

regression

model.Thequadratic

regressionmodel had an

average errorpercent of

about 1.36 while


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the others had

highererror.

The only modelthat came close

was the linearregression

model with anerror


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percentage of

1.53.Predictions

These

predictions aregoing to be with

the model thatI thought was

the best.


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Quadratic

Regressionmode

l. The equationis:


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1984

Predictions

2016PredictionsFirst

we mustsubstitute in

the year for xand thenwe can

find y which is


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0 1 9 3 0 1 9 4

0 1 9 5 0 1 9 6

0 1 9 7 0 1 9 8

0 1 9 9 0 2 0 0

0 2 0 1 0 2 0 2

0 2 0 3 0 He i g h t ( c m )Year


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Year Of

Olympics

VS. GoldMedalH


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eightsQuadraticRegression


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Height in cm =

239.75

295.8Table:X-ValuesInitialval

uesQuadraticRegression1932

197197.995296519

36 203


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199.0379511194

8 198

204.33482091952 204

206.82341271956 212

209.67348881960 216

212.885049319


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64 218

216.458094196

8 224220.392623197

2 223224.688636419

76 225229.346134198


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0 236

234.36511591984239.752016

295.8

Graph:0501001502002503001 8 8

0 1 9 0 0


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1 9 2 0 1

9 4 0 1 9

6 0 1 9 8

0 2 0 0 0

2 0 2 0 He i g h t ( c m )Year


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Year Of

Olympics

VS. GoldMedalH


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eightsInitialvaluesQuadraticRegressionMy answers for1984 make

sense becauseit is not too far


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from the other

values like that

of 1980. Ontheother hand

my value for2016 is high.

The mainproblem with

any of the


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models is that

they show

thatthe heightis continuously

rising. It is veryunlikely that

the resultswould keep

rising like the


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models

Isuggested.

Since 2016 farfrom the years

given we cansafely say that

neither of themodels

stated(linear or


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quadratic) are a

suitable

regression touse.Additional

Data

a)Year 1896

1904 1908 1912


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1920 1928 1984

1988 1992 1996

2000 20042008Height(cm

)190 180 191193 193 194

235 238 234239 235 236

236


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0501001502002

503001 8 8

0 1 9 0 0

1 9 2 0 1

9 4 0 1 9

6 0 1 9 80 2 0 0 0

2 0 2 0 He i g h t ( c m )


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YearYear Of

Olympics

VS. Gold

MedalH


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eightsInitialvaluesQuadraticRegressionb)As we can seethe model does

fit the newdata for the


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most part. The

only part that

we see notfittingthe data

is when thequadratic

regressionstarts

increasing from


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1984 and

beyond.Year

1896 1904 19081912 1920 1928

1932 1936 19481952

1956Height(cm)190 180 191 193

193 194 197


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203 198 204

212Year 1960

1964 1968 19721976 1980 1984

1988 1992 19962000Height(cm

)216 218 224223 225 236

235 238 234


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239 235Year

2004

2008Height(cm)236 236As we

can see fromthe graph the

quadraticregression

seems to follow


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the data

closely. There

are onlysomevalues at the

beginning and atthe end where

the quadraticregression is

different. From


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the tablebelow

we can see that

most of thevalues are

reasonablyclose to the

given actualvalue. A thing

thatcould be


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done to make

the data more

accurate ismake a new

quadraticregression line

for all ofthedata and not


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just the initial

data given.X-ValuesInitialval

uesQuadraticRegression1896

190204.878198190

4 180


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200.818274319

08 191

199.33053881912 193

198.20428771920 193

197.03623831928 194

197.3141261193


258/306

2 197

197.995296519

36 203199.0379511194

8 198204.334820919

52 204206.823412719

56 212


259/306

209.673488819

60 216

212.88504931964 218

216.4580941968 224

220.3926231972 223

224.688636419


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76 225

229.346134198

0 236234.365115919

84 235239.745582119

88 238245.487532719

92 234


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251.590967519

96 239

258.05588662000 235

264.882292004 236

272.07017772008 236

279.6195497


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Further testingand applicationThe patterns

that showed up

Jump also show

up in other

sports in the


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Olympics.

Forexample the

Olympicrecords for

free style

shows this samepattern.Year

1896 1904 1908


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1912 1920 1928

1932 1936 1948

1952 1956Time(seconds)82.2

62.8 65.6 63.461.4 58.6 58.2

57.6 57.3 57.455.4Year 1960

1964 1968 1972


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1976 1980 1984

1988 1992 1996

2000Time(seconds)55.2

53.4 52.2 51.249.9 50.4 49.8

48.6 49.0 48.748.3Year 2004


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2008Height(cm

)48.2 47.20102030405060

7080901 8

8 0 1 9 0

0 1 9 2 0

1 9 4 0 1

9 6 0 1 98 0 2 0


267/306

0 0 2 0

2 0 He i g h t ( c m )YearYear Of

Olympics


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VS. Gold

Medal

HeightsInitial values

When the data

is graphed it


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looks like

this:The X

valuesrepresent the

years in whichthe Olympics

were hostedinThe Y values

represent the


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time of the gold

medalist

resultsThegraph shows a

decline in timesince the early

1900s. The gold

results were


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becomingshorte

r and shorter

over the yearuntil towards

the start of the21stcentury. Like on

the Gold


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medalistheights

this suggests a

type ofasymptote. If

we do aquadratic

regression forthis data we

get theequation


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f(x) =

.0017830124x^

2-7.173749547x+

7264.004005.When we graph

this equation weget


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thefollowing

graph.0102030405060

7080901 8

8 0 1 9

0 0 1 9

2 0 1 9

4 0 1 96 0 1 9


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8 0 2 0

0 0 2 0


Year Of

Olympics


276/306

VS. Time

of 100m

FreeStyleInitial


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X-Values Initial

values

QuadraticRegression1896

82.272.1763676119

04 62.868.9899481919

08 65.6


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67.4823230819

12 63.4

66.031754361920 61.4

63.301786121928 58.6

60.800043471932 58.2

59.6347567319


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36 57.6

58.5265264194

8 57.355.5441737719

52 57.454.6641690319

56 55.453.8412206719

60 55.2


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53.0753287219

64 53.4

52.366493161968 52.2

51.714714197251.2

51.119991241976 49.9

50.5823248719


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80 50.4

50.10171491984

49.849.6781613319

88 48.649.31166415199

2 4949.0022233719


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96 48.7

48.749838990102030405060

7080901 8

8 0 1 9

0 0 1 9

2 0 1 9

4 0 1 96 0 1 9


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8 0 2 0

0 0 2 0


Year Of

Olympics


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VS. Time

of 100m

Freestyle2000 48.348.5545112004

48.2


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48.4162394120

08 47.2

48.33502422Compare and

contrast:


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0501001502002

503001 8

8 0 1 9

0 0 1 9

2 0 1 9

4 0 1 96 0 1 9

8 0 2 0


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0 0 2 0

2 0 He i g h t ( c m )YearYear Of

Olympics


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VS. Gold

Medal

HeightsFrom the twographs above

we can see that


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they are really

opposites of

each other. Thegold

medalheightsare a concave

up quadraticfunction while

the time for


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freestyle is

concavedown.The gold medal

heights graph isincreasing

throughout thedata and the


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freestyle

isdecreasing

throughout thedata. They are

both similar inthe way that

accurately

model


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everysingle

year that the

Olympics will beheld because

they will bothdecrease and

increase beyondtheactual

values. This is


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because humans

are competing

in these eventsand we all have

limits. This isalsoapparent in

the gold medalheights, as the

years went on


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the heights got

more and more

close toeachother

withoutsignificant

difference. The

freestyle seems


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to fit the data

better

becausetowardsrecent years

the times havebeen very close.

Over all we cansee that the


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freestylequadra

tic regression

model fits itsdata better

than the goldmedal heights.Conclusion

a)


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jump results

from 1896 to2008 Olympics

showed thatthe gold

medalistresultsfor high

jump steadily


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increased.

Towards the

end the heightsstarted to level

off because ofhuman limits.

The best modelthat was found

to model the


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data was

quadratic. This

is becausefrom896 towards

2008 the datawas steadily

increasing at aparabola like

shape. The


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m

Freestyle

results from1896 to 2008

showed thatthe results for

the 100 mfreestyle

weresteadily


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decreasing, and

towards the

end the resultsleveled off.

This is becauseof

humanlimitations. A good type

of model to


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model the 100m

freestyle data

from 1896 to2008 was alsoa

quadraticfunction. This

modeled thedata almost

perfectly


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BibliographySwim-City.

"Swim-City.com

- Record


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History Olympic

Records Men."Swim-City.com -SwimmingMetro

polis. Swim-City,

2011. Web. 20Jan. 2012.


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Documents

Math Gold Medal