ia mathematics sl volume by revolution

8/18/2019 IA Mathematics SL Volume by revolution

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Candidate: 000738-0003 Arandia Jimenez 1

Mathematics SL

Internal Assessment

Calculating the Volume of a Designed Flower Vase

Candidate: Maria René Arandia Jiménez

Candidate Number: 000738-0003

Session: May 201

Course: !" Mat#emati$s S%

I. Introduction

Ca&$u&us is t#e study o' #o( t#in)s $#an)e* by (#i$# a +erson $an $reate mode&s and

+redi$t ,uantitatie $#an)es and se,uen$es. /#at !s Ca&$u&us and #y do (e Study it4

5i''erentiation and !nte)ration are t#e t(o main bran$#es o' Ca&$u&us. ! (as attra$ted to

!nte)ra&s6 ! sa( it as a se$ond met#od o' 'indin) t#e o&ume o' an obe$t* and not on&y


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mu&ti+&yin) t#e dimensions o' a )eometri$a& t#ree-dimensiona& s#a+e. Considerin) its +ur+ose* !

(ondered: Cou&d be +ossib&e to $a&$u&ate t#e o&ume o' non-re)u&ar s#a+es* su$# as a ase

i)ure 1. s#o(s a sine 'un$tion on a )ra+# (it# its s#aded area under t#e &ine* and i)ure 2.

s#o(s #o( it &oo9s (#en t#e area is rotated 3 around t#e -ais.0

i)ure 1. and 26 So&id o' Reo&ution - indin) ;o&ume by Rotation6 yzAnt6 (yzant.$om6

eb6 10 Mar$#* 201.

r is $#an)in) 'or eery +oint in t#e -ais* and i' r ¿ f ( x )= y * t#e area o' t#e ima)inary

$ross se$tiona& $ir$&e inside t#e ase (i&& a&so be $#an)in)* be$ause it is an irre)u&ar 35 s#a+e. !n

order to 'ind t#e o&ume o' t#e ase usin) t#e area ( A=π r2 ¿ o' its ima)inary $ross se$tiona&

dis9s inside* t#e o&ume o' a $ross se$tiona& dis9 #as to be $a&$u&ated.


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! 'e&t ent#usiasti$ to 'ind t#e o&ume o' an obe$t* but to $#a&&en)e myse&'* ! de$ided to

$reate a '&o(er ase and mode& it. #i&e desi)nin) t#e ase* ! )ot ins+ired by t#e >o&den ratio*

(#i$# are used 'or art(or9s and ar$#ite$ture sin$e An$ient >ree$e* and is 'ound in many as+e$ts

o' nature* su$# as '&o(ers. A >o&den s+ira& $reated by t#e >o&den re$tan)&e (i&& be 'an$y?

i)ure 36 >o&den S+ira&6 o&'ram Mat# or&d6 mat#(or&d.(o&'ram.$om6 eb6 1@ Mar$#*

201.

o&den ratio φ is to be obtained t#rou)# t#e diision o' &en)t# /%4 o' t#e re$tan)&e

by its (idt# /4:

Goldenratio=φ ≈ L

W

o&den s+ira&

φ


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Candidate: 000738-0003 Arandia Jimenez

φ=1+√ 5

2 /usin) a 22 $m s,uare4 ≈1.6180339887….

% (i&& be $a&$u&ated by addin) #a&' on t#e s,uareBs base and t#e &en)t# o' t#e red

dia)ona&:

L= x

2+√ x2+(

x

2)2

! (i&& use t#is met#od to 'ind t#e distan$e bet(een t#e +oints o' t#e 'un$tions t#at 'o&&o(

t#e )o&den s+ira& s#a+e* so ! $an mode& t#e ase and t#en 'ind a best-'it 'un$tion t#at (i&& a&&o(

me to 'ind its o&ume. ina&&y* ! (i&& rotate t#e area to 'ind t#e o&ume.

II. Aims

!nte)ra&s )ie us #umans a met#od to 'ind t#e o&ume o' ori)ina& and non-$onentiona&

obe$ts.


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Candidate: 000738-0003 Arandia Jimenez @

III. Modeling

!n i)ure . ! #ad to de'ine t#e +oints t#at ! (anted to )ra+#. indin) t#e +oints (i&& be

easy 'or bot# o' t#e &ines /be'ore t#e ye&&o( dots4* but 'rom t#e ye&&o( dots to t#e ri)#t* t#e &ines

'o&&o( t#e )o&den ratio +ro+ortions* so ! #ad to de$ide (#at dimensions t#e s,uare (i&& #ae* inorder to 'ind t#e ot#er +oints on t#e )o&den s+ira&.

! de$ided t#at t#e dimensions o' t#e s,uares (i&& be $m 'or t#e red s+ira& /A"C54 and

2.@ 2.@$m 'or t#e b&a$9 s+ira& /D>E4. ! (i&& re'er to t#e &en)t# as t#e &on)er side o' t#e

re$tan)&e* and (idt# as t#e s#orter side.

Red S+ira& /A"C54

As s#o(n in i)ure * t#e )reen +oints #ae to be &o$ated on t#e $oordinate +&ane. oint

A on t#e red s+ira& is a&ready de'ined as /1.F* .74* so it is ne$essary to 'ind t#e +oint ". !t is

obious t#at +oint " (i&& be at /18.F* 8.74 be$ause o' t#e estab&is#ed dimensions o' t#e s,uare

/$m to t#e ri)#t and $m u+(ards4.


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Candidate: 000738-0003 Arandia Jimenez


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o&den ratio in Red s+ira&

Re$tan)&e % /$m4 /$m4φ=

L

W

1 R 2+2√ 5 1+√ 52

2 R 2√ 5−2 1+√ 52

3 R 2√ 5−2 6−2√ 5 1+√ 52

roo':


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Candidate: 000738-0003 Arandia Jimenez F

L R1

W R1=φ

4

2√ 5−2=

1+√ 5

2

2

√ 5−1∙(√

5+1√ 5+1

)=1+√ 52

2√ 5+24

=1+√ 52

√ 5+12

=1+√ 52

"&a$9 S+ira& /D>E4

! used t#e ea$t same +ro$edure to 'ind t#e $oordinates o' t#e b&a$9 s+ira& /D>E4.

o&den ratio.

$a%le 2. 5imensions o' "&a$9 re$tan)&e and +roo' o' >o&den ratio in "&a$9 s+ira&

Re$tan)&es % /$m4 /$m4φ=

L

W

1B 5+5√ 54

2.@ 1+√ 52

2B 2.@ −5+5√ 54

1+√ 52

I


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3B −5+5√ 54

15−5√ 54

1+√ 52

roo':

L R1W R1

=φ

(−5+5√ 5

4)

(15−5√ 5

4)=

1+√ 52

−1+√ 53−√ 5 ∙(

3+√ 53+√ 5 )=

1+√ 52

2√ 5+24

=1+√ 52

√ 5+12

=1+√ 52

Figure &. >ra+# s#o(in) set o' 'un$tions and its $oordinate +oints

I


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S1

==++

=+++

=+++

=+++

[email protected]@/4.1@/

[email protected]@.12/[email protected]/

2.8F.74F.7/4F.7/

8F.A4F.A/4F.A/

23

23

23

23

d cba

d cba

d cba

d cba

(#i$# (i&& be$ome:

S1=¿

=+++

=+++

=+++

=+++

[email protected]

[email protected]@.1@[email protected]@3

2.8F.7A1.203F.AF3

8F.A01.2AAF.117

d cba

d cba

d cba

d cba

! (i&& so&e t#e system o' e,uations by e&imination:

Subtra$t 1 'rom 2 to e&iminate d:

¿ 493.039a+62.41b+7.9c+d=8.2117.649a+24.01b+4.9c+d=8375.39a+38.4b+3c=0.2 A

Subtra$t 3 'rom to e&iminate d:

¿3796.416a+243.366b+15.6c+d=6.61953.125a+156.256b+12.5c+d=7.11843.291a+87.11b+3.1c=−0.5B

Subtra$t 2 'rom 3 to e&iminate d:

¿1953.125a+156.256b+12.5 c+d=7.1

493.039a+62.41b+7.9c+d=8.21460.086a+93.846b+4.6c=−1.1C

Subtra$t A 'rom B to e&iminate $:

1

2

3


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¿1843.291a+87.11b+3.1c=−0.5(¿3)375.39a+38.4b+3c=0.2(¿3.1)

¿5529.873a+261.33b+9.3c=−1.5

1163.709a+119.04b+9.3c=0.624366.164a+142.296b=−2.12

Subtra$t A 'rom C to e&iminate $:

¿1460.086a+93.846b+4.6c=−1.1(¿3)375.39a+38.4b+3c=0.2(¿ 4.6)

¿ 4380.258a+281.538b+13.8 c=−3.31726.794a+176.64b+13.8c=0.922653.464a+104.898b=−4.22

Subtra$t I 'rom II to e&iminate b:

¿2653.464a+104.898b=−4.22(¿142.296)4366.164 a+142.296b=−2.12(¿104.898)

¿377577.3133

a+14926.56581

b=−600.48912

458001.8713 a+14926.56581b=−222.38376−80424.558a=−378.10536

a=−378.10536−80424.558

a≈4.7014 K 10−3

Re+&a$e a in I to 'ind b:

4.7014

4366.164 ¿ K 10−3¿+142.296b=−2.12

b≈−0.1592


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Re+&a$e a and b in A to 'ind $:

4.7014

375.39 ¿ K 10−3¿+38.4(−0.1592)+3c=0.2

c ≈1.5161

Re+&a$e a* b* and $ in 1

4.7014

117.649¿ K 10−3 ¿+24.01(−0.1592)+4.9(1.5161)+d=8

d ≈3.8404

un$tion '/4 resu&ts in:

y=4.7014 K 10−3

x3−0.1592 x2+1.5161 x+3.8404

Equation test:

/.F* 84: Re+&a$e .F on '/4:

y=4.7014 K 10−3(4.9)3−0.1592 (4.9 )2+1.5161(4.9)+3.8404

y ≈8.0000

/7.F* 8.24: Re+&a$e 7.F on '/4:

y=4.7014 K 10−3

(7.9)3

−0.1592 (7.9 )2

+1.5161(7.9)+3.8404

y ≈8.1999

/1@.* .4: Re+&a$e 1@. on '/4:


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Candidate: 000738-0003 Arandia Jimenez 1@

y=4.7014 K 10−3(15.6)3−0.1592 (15.6 )2+1.5161(15.6)+3.8404

y ≈6.5971

Figure . >ra+# o' manua& and automati$ $ure 'it o' '/4.


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y ≈6.7800

/17* 7.@4: Re+&a$e 17 on )/4:

y=0.3829(17)3

−19.2724 (17)2

+323.9507(17)−1811.3755

y=7.2505

/17.* 8.14: Re+&a$e 17. on )/4:

y=0.3829(17.6)3−19.2724 (17.6 )2+323.9507(17.6)−1811.3755

y=7.8232

Figure 1/. >ra+# o' manua& and automati$ $ure 'it o' )/4


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A'ter t#e e,uation test* ! )ra+#ed t#e +oints on %o))er ro +a$9a)e +ro)ram /+in9 &ine4

and obsered t#at t#e auto 'it $ure (as more a$$urate t#an t#e manua& 'it $ure /b&ue &ine4

be$ause it +assed t#rou)# a&& t#e +oints o' t#e 'un$tion )/4. A)ain* ! de$ided t#at t#e automati$

$ure 'it (i&& be used in order #ae a more +re$ise $a&$u&ation o' t#e o&ume.

un$tion 3 /#/44

or 'un$tion #/4* a $ubi$ mode& (i&& be used:

y=ax3+b x2+cx+d


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Candidate: 000738-0003 Arandia Jimenez 1F

and re+&a$ed t#e ariab&es in t#e $ubi$ e,uation* and t#e resu&tin) 'un$tion #/4 is:

y=0.0842 x3−5.1127 x2+100.1224 x−627.4719

Equation test:

/17.2* 10.34: Re+&a$e 17.2 on #/4:

y=0.0842(17.2)3−5.1127 (17.2 )2+100.1224 (17.2)−627.4719

y ≈10.5395

/1.@@@* 10.@4: Re+&a$e 1.@@@ on #/4:

y=0.0842(16.555)3−5.1127 (16.555 )2+100.1224 (16.555)−627.4719

y ≈10.8588

/[email protected]* F.F04: Re+&a$e [email protected] #/4:

y=0.0842(15.9)3−5.1127 (15.9 )2+100.1224 (15.9)−627.4719

y ≈10.3895

Figure 11. >ra+# o' manua& and automati$ $ure 'it o' #/4.


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"ased on (#at t#e )ra+# +roides* i/4 $an be:

i ( x )=0.01573 x3−0.4696 x2+4.120 x−4.382

un$tion @ //44

Figure 13. >ra+# o' manua& and automati$ $ure 'it o' /4.

or 'un$tion /4* a $ubi$ mode& (i&& be used* ta9in) t#e 'orm o':

# ( x )=0.1551 x3−7.585 x2+123.8 x−669.1

un$tion /9/44

Figure 1. >ra+# o' manua& and automati$ $ure 'it o' 9/4.


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A $ubi$ mode& (i&& be used 'or 'un$tion 9/4* and (it# %o))er ro mode&* it (i&& resu&t:

$ ( x )=−0.005484 x3−0.2414 x2+12.61 x−106.3


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Stage 1:

Figure 1!. ;o&ume 1 / V

1 4 /s#aded ye&&o( area4 t#at is to be 'ound in Sta)e 1

Re+&a$e '/4 in e,uation:

V f ( x) = ∫b

a

π (0.004314 x3−0.1469 x2+1.395 x+4.198)2 dx

V f ( x) =

∫0

15.6

π ((1.8611∗10−5 ) x6

−(1.2675∗10−3) x5+0.0336 x4−0.3736 x3+0.7126 x2+11.7124 x+17.6232)dx

V f ( x)=[ π (1.8611∗10−5

7 x

7−1.2675∗10−3

6 x

6+0.0336

5 x

5−0.3736

4 x

4+0.7126

3 x

3+11.7124

2 x

2+17.6232 x

V f ( x)≈2613.7947 c'3

or t#e 'o&&o(in) resu&ts o' t#e inte)rated 'un$tions* Casio '-F80>!! S5 )ra+#i$s

$a&$u&ator (as used to 'ind t#e o&ume* (it# de$ima& +&a$es.

Re+&a$e )/4 in e,uation:


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V ! ( x) = ∫15.6

18.1

π (0.1866 x3−9.113 x2+148.9 x−806.9)2dx

V ! ( x)≈452.1702 c'3

Re+&a$e i/4 in e,uation:

V i( x) = ∫

1.4

14.9

π (0.01573 x3−0.4696 x2+4.120 x−4.382)2dx

V i( x)≈1239.1658 c'3

Re+&a$e /4 in e,uation /&imit: a 18.14:

V # ( x) = ∫

14.9

18.1

π (0.1551 x3−7.585 x2+123.8 x−669.1)2 dx

V # ( x)≈293.7897 c'3

Add "&a$9 s#a+e o&umes and Red s#a+e o&umes* se+arate&y.


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Candidate: 000738-0003 Arandia Jimenez 2@

Re+&a$e 9/4 in e,uation:

V $ ( x) = ∫14.9

18.9

π (−0.005484 x3−0.2414 x2+12.61 x−106.3)2 dx

V $ ( x)≈1444.6748 c'3

Re+&a$e #/4 in e,uation:

V "( x) = ∫

15.6

18.1

π (0.1630 x3−8.989 x2+163.6 x−973.5)2dx

V "( x)≈850.5552 c'3

Subtra$t to 'indV

2 :

V 2=V

$ ( x )−V " ( x )1444.6748−850.5552

V 2=594.1196 c'3

As s#o(n in i)ure 10* t#ere are t(o +in9 s#aded areas /! and !!4 t#at are not subtra$ted

'romV

$ ( x ) * soV

2 (i&& not be a$$urate.


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( x )=2330

x−2.27 ( x )=− x2+18.15

V ( x)=∫14.9

15.6

π (23

30 x−2.27)

2

dx V ( x)=∫18.1

18.9

π (− x2+18.15)

2

dx

V ( x)=195.2634 c'3

V ( x)=199.1200 c'3

V (inal2=V 2−(V ( x )+V ( x ))

V (inal2=199.7362c'3

Stage 3:

Figure 1#. ;o&ume 3 / V

3 4 /s#aded ye&&o( area4 t#at is to be 'ound in Sta)e 3

Re+&a$e /4 / 18.1& x &18.9 4 in e,uation:

V # ( x) = ∫18.1

18.9

π (0.1551 x3−7.585 x2+123.8 x−669.1)2 dx

V # ( x)≈135.5808 c'3

Subtra$tV # ( x) 'rom

V ( x) to )et ;o&ume 3:


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V 3=V

( x )−V #( x )

V 3=63.5392 c'3


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by seera& industries* (#i$# )ies us anot#er reason to +roo' t#at $a&$u&us is ery im+ortant in

rea& &i'e.

VI. 0eferences

•

C&i+ 2: So&ids o' Reo&ution. Session @7: Eo( to Ca&$u&ate ;o&umes. Massa$#usetts!nstitute o' o&den S+ira&. 5i)ita& ima)e. o&'ram Mat# or&d. N.+.* n.d. eb. 10

Mar. 201. Qmat#(or&d.(o&'ram.$om.

ro)rams and too&s used in t#e e+&oration:

• Casio '-F80>!! S5 )ra+#i$s $a&$u&ator

• Mi$roso't ord

• %o))er ro )ra+#in) so't(are• aint )ra+#i$s +ro)ram
http://ocw.mit.edu/http://ocw.mit.edu/http://ocw.mit.edu/

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