Part B – Study of the perimeter

Homework #2

In this exerise,we study Koh's snowake, agure with surprising properties.

Part A – Construction of the figure

This onstrution an bedone by hand orusing a software suh asGeogebra. Draweah

new gure as anew one, and not onthe previous one.

ToreateaKohsnowake,startwithanequilateraltriangleofside

9

^{m,thenreursively}

alter eah linesegment asfollows :

1. Divide the linesegmentintothree segmentsof equallength.

2. Draw an equilateral triangle that has the middle segment from step 1 as its base

and pointsoutward.

3. Remove the line segment that isthe base of the trianglefrom step 2.

This proessantheoretially goonadinnitum,butyouwillstopaftertworeplaement

steps.Inthisway,youshouldendupwith

3

^{dierentgures,inludingthe initialtriangle.}

Part B – Study of the perimeter

Weall

P ( n )

^{theperimeterofthegureafter}

n

steps.Forexample,

P (0)

^{istheperimeter}

of the initialtriangle, and

P (2)

^{is the perimeter of the gure after}

2

^{steps, the lastone}

youdrew.

1. Compute

P (0)

^{, the perimeterof the initialtriangle.Explain your} omputation.

2. Compute

P (1)

^{justifying every part of your} omputation.

3. Compute

P (2)

^{justifying every part of your} omputation.

4. Compute the ratios

P (1) P (0)

^and

P (2)

P (1)

^{. What do you notie?}

5. Prove that eahreplaementstep multipliesthe perimeterof the gure by

4 3

^.

6. Dedue the perimeters of

P (3)

^,

P (4)

^,

P (5)

^and

P (6)

^{. Give the answers asexat}

values and rounded values to 3DP.

7. Use your alulator tond a value of

n

suh that

P ( n ) > 250

^.

8. Whatanyousay about

P ( n )

^when

n

getsgreaterandgreater?Doyouthinkthere is a maximumvalue for

P ( n )

^?

Part C – Study of the area

We all

A ( n )

^{the area of the gure after}

n

steps. For example,

A (0)

^{is the area of the}

initialtriangle,and

A (2)

^{is the area of the gure after}

2

^{steps, the last one you drew.}

1. Find the formula for the area of an equilateral triangle of side

s

m. Explain your method.

2. Use the formula to ompute

A (0)

^{, the area of the initial triangle. Give the exat}

value and arounded value to3 DP.

3. Use the formula to ompute the area of an equilateral triangle of side

3

^{m and}

dedue the value of

A (1)

^{. Givethe exat value and a rounded value to 3DP.}

4. Use the formula to ompute the area of an equilateral triangle of side

1

^{m and}

dedue the value of

A (2)

^{. Givethe exat values and rounded values to3 DP.}

5. Compute the dierenes

A (1) − A (0)

^and

A (2) − A (1)

^{.Givethe exat valueand}

a rounded value to 3DP. What do you notie?

6. Do youthink the area of the gure willgrow like itsperimeter?

7. Find onthe interneta proof of the fat that the area of the gure stays nite.

8. What is the most surprising feature of about Koh's snowake?