IA Mathematics SL Volume by revolution
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Transcript of 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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Candidate: 000738-0003 Arandia Jimenez 3
! '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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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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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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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 $oordinate +oints
I
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Candidate: 000738-0003 Arandia Jimenez 12
S1
==++
=+++
=+++
=+++
[email protected]@/4.1@/
[email protected]@.12/[email protected]/
2.8F.74F.7/4F.7/
8F.A4F.A/4F.A/
23
23
23
23
d cba
d cba
d cba
d cba
(#i$# (i&& be$ome:
S1=¿
=+++
=+++
=+++
=+++
[email protected]@.1@[email protected]@3
2.8F.7A1.203F.AF3
8F.A01.2AAF.117
d cba
d cba
d cba
d cba
! (i&& so&e t#e system o' e,uations by e&imination:
Subtra$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$e a in I to 'ind b:
4.7014
4366.164 ¿ K 10−3¿+142.296b=−2.12
b≈−0.1592
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Candidate: 000738-0003 Arandia Jimenez 1
Re+&a$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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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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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/