13. Surface Area and Volume

8/12/2019 13. Surface Area and Volume

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R N K ^ F L H \ E K F B UT - C

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]^RCHMN HRNH] HFL SKJÎN]

EF\RKL^M\EKFEf tès m`hptnr, s`hjj lesmuss prkbjnis kf mkfvnrsekf kc kfn kc t`n skjels jegn mubkel, mubn, rea`t mermujhr myjeflnrrea`t mermujhr mkfn hfl sp`nrn ef hfkt`nr.Ef kur lhy-tk-lhy jecn wn mkin hmrkss vhrekus skjels wèm` hrn mkibefhtekfs kc twk kr ikrn sum` skjels, Cknxhipjn, h mkfemhj mermus tnft wet` myjeflremhj bhsn es h mkibefhtekf kc h rea`t mermujhr myjeflnr hfl h rea`t mermujhmkfn, hjsk hf emn-mrnhi mkfn es h mkibefhtekf kc h mkfn hfl h `nie-sp`nrn. Qn s`hjj lesmuss prkbjnis kf ceflefsurchmn hrnhs hfl vkjuins kc sum` skjels. Qn hjsk mkin hmrkss skjels wèm` hrn h phrt kc h mkfn. Ckr nxhipjn, bumgnt , h ajhss tuibjnr, h cremtekf mjutm` ntm. t`nsn skjels hrn gfkwf hs crustuis kc h mkfn. Ef t`n nfl kc t`m`hptnr, wn s`hjj lesmuss prkbjnis kf surchmn hrnh hfl vkjuin kc crustui kc h mkfn.

^FE\] KC INH]^RNINF\ KC HRNH HFL SKJÎN\`n eftnr-rnjhtekfsèps bntwnnf vhrekus ufets kc inhsurninft kc jnfat`, hrnh hfl vkjuin hrn jestnl bnjkw ckr rnhlyrncnrnfmn?

JNFA\@

8 Mnfteintrn (mi) 9 84 iejeintrn (ii)8 Lnmeintrn (li) 9 84 mnfteintrn

8 Intrn (i) 9 84 li 9 844 mi 9 8444ii8 Lnmhintrn (lhi) 9 84 i 9 8444 mi8 @nmtkintrn (ì) 9 84 lhi 9 844 i8 Gejkintrn (gi) 9 8444 i 9 lhi 9 84 ì8 Iyrehintrn 9 84 gejkmntrn

HRNH

8 mi29 8 mi x 8 mi 9 84 ii x 84 ii 9 844 ii28 li29 8 li x8 li 9 84 mi x84 mi 9 844 mi28 i29 8 i x8 i 9 84 li x84 li 9 844 li28 lhi29 8 lhi x8 lhi 9 84 i x84 i 9 844 i28 ì28 nmthrn 9 8 i x8 ì 9 844 i x844 i 9 84444 i29 844 li2

8 gi2

9 8 gi x8gi 9 84 ì x84 ì 9 844 ì2

kr 844 `nmthrn

SKJÎN8 mi= 9 8 ij 9 8 mi x8 mi x8mi 9 84 ii x84 ii x84 ii 9 8444 ii=8 jetrn 9 8444 ij 9 8444 mi=8 i= 9 8 i x8 i x8 i 9 844 mi x844 mi x844 mi 9 843mi=9 8444 jetrn 9 8 gejkjetrn8li= 9 8444 mi=8 i= 9 8444 li=8 gi= 9 845i=

M^BKELH rnmthfaujhr skjel bkuflnl by sex rnmthfaujhr pjhfn chmns es mhjjnl h mubkel.H ihtm` bkx, h tnh-phmgnt, hbremg, h bkkg, ntm., hrn hjj nxhipjns kc h mubkel.H mubkel `hs 3 rnmthfaujhr chmns, 82 nlans hfl 0 vnrtemns.


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\`n ckjjkwefa hrn skin lncefetekfs kc tnris rnjhtnl tk h mubkel.(e) \`n sphmn nfmjksnl by h mubkel es mhjjnl ets vkjuin.(ee) \`n jefn okefefa kppksetn mkinrs kc h mubkel es mhjjnl ets lehakfhj.

H mubkel `hs ckur lehakfhjs.H lehakfhj kc h mubkel es t`n jnfat` kc t`n jkfanst rkl t`ht mhf bn pjhmnl ef t`n mubkel.

(eee) \`n sui kc hrnhs kc hjj t`n sex chmns kc h mubkel es gfkwf hs ets tkthj surchmn hrnh.(ev) \`n ckur chmns w`em` innt t`n bhsn kc h mubkel hrn mhjjnl t`n jhtnrhj chmnskc t`n mubkel.(v) \`n sui kc hrnhs kc t`n ckur whjjs kc h mubkel es mhjjnl ets jhtnrhj surchmn hrnh.

Ckriujhn

Ckr h mubkel kc jnfat` 9 ufets, brnhlt` 9 b ufets hfl `nea`t 9 ` ufets, wn `hvn?

]ui kc jnfat`s kc hjj nlans 9 < )( `b ufets.

Lehakfhj kc mubkel 9222

`b ufets.\kthj ]urchmn Hrnh kc mubkel 9 2 )( `b`b sq. ufets.Jhtnrhj ]urchmn Hrnh kc mubkel 9 W2 `b )( Y sq. ufets.Hrnh kc ckur whjjs kc h rkki 9 W2 `b )( Y sq. ufets.

Skjuin kc mubkel 9 )( `b mubem ufets.

RNIHRG? Ckr t`n mhjmujhtekf kc surchmn hrnh, vkjuin ntm. kc h mubkel, t`n jnfat`, brnhlt` hfl `nea`t iustbn nxprnssnl ef t`n shin ufets.

M^BN

H mubkel w`ksn jnfat`, brnhlt` hfl `nea`t hrn hjj nquhj

es mhjjnl h mubn .

Emn-mubns, ]uahr, Lemn, ntm. hrn hjj nxhipjns kc h mubn.

Nhm` nlan kc h mubn es mhjjnl ets seln.

Ckriujhn

Ckr h mubn kc nlan 9 h ufets, wn `hvn7

]ui kc jnfat` kc hjj nlans 9 82 h fuets.

Lehakfhj kc mubn 9 )=(h fuets.\kthj ]urchmn Hrnh kc mubn 9 (3h2) sq. fuets.Jnhtnrhj ]urchmn Hrnh kc mubn 9 (


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Nx.8 \`n jnfat`, brnhlt` hfl nea`t kc h rnmthfaujhr skjel hrn rhtek 3 ? > ? =20 mi2,cefl t`n jnfat`, brnhlt` hfl `nea`t kc t`n skjel.

]kj. Jntjnfat`9 (3x) mi, brnhlt` 9 (>x) mi hfl `nea`t 9 (x+>xx mi brkhl hfl 54 mi lnnp.

Cefl? (e) t`n mhphmety kc t`n mestnrf ef jetrns?(ee) t`n vkjuin kc erkf usnl7(eee) t`n tkthj surchmn hrnh kc t`n mestnrf.

]kj. Nxtnrfhj leinfsekfs kc t`n mestnrf hrn ?

Jnfat` 9 82> mi, Brnhlt` 9 84> mi hfl Lnpt` 9 54 mi.Eftnrfhj leinfsekfs kc t`n mestnrf hrn ?Jnfat` 9 824 mi, Brnhlt` 9 844 mi hfl Lnpt` 9 01.> mi.

(e) Mhphmety 9 Eftnrfhj vkjuin 9 (824x844x01.>) mi=9

8444

>.01844824jetrns 9 84>4 jetrns.

(ee) Skjuin kc erkf 9 (Nxtnrfhj vkjuin) (Eftnrfhj vkjuin) 9 W(82>x84>x54) (824x844x01.>)Y mi=9 (88082>4 84>4444) mi=9 8=82>4 mi=.

(eee) Nxtnrfhj hrnh 9 (Hrnh kc < chmns) + (Hrnh kc t`n bhsn) 9 (W2(82> + 84>) x54Y + (82> x84>)} mi2.

9 (2> mi2.Eftnrfhj hrnh 9 (W2(824 + 844)x01.>Y + (824x844)} mi29 (=0>44 + 82444) mi29 >4>44 mi2.Hrnh ht t`n tkp 9 Hrnh bntwnnf kutnr hfl effnr rnmthfajns 9 W(82> x84>) (824 x844)Y mi29 (8=82> 82444) mi29 882> mi2.

\kthj surchmn hrnh 9 (>2> + >4>44 + 882>) mi29 8438>4 mi2.

Nx.=. H cenjl es 04 i jkfa hfl >4 i brkhl. Ef kfn mkrfnr kc t`n cenjl, h pet w`em` es 84 i jkfa, 1.> i brkhl hfl 0 ilnnp `hs bnnf lua kut. \`n nhrt` thgnf kut kc et es nvnfjy sprnhl kvnr t`n rnihefefa phrt kc t`n cenjl. Ceflt`n resn ef t`n jnvnj kc t`n cenjl.

]kj. Hrnh kc t`n cenjl 9 (04 x>4) i29 ) i29 8> i2

Hrnh kvnr w`em` t`n nhrt` es sprnhl kut 9 () i2 9 =52> i2

Skjuin kc nhrt` lua kut 9 (84 x1.> 0) i=9 344 i=.

Resn ef jnvnj 9 mimiiHrnh

vkjuin=.8>

=52>

844344

=52>

344


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i lnnp. Ef w`ht tein mhf t`n whtnr bniptenl by h pepn kc w`em` t`n mrkss snmtekf es h squhrn kc seln 24 mi, ec whtnr rufs t`rkua` t`n pepn ht t`nrhtn kc 8> gi/ `r ;

]kj. Skjuin kc whtnr ef t`n rnsnrvker 9 (04x34x3.>) i=9 =8244 i=.

Hrnh kc mrkss snmtekf kc t`n pepn 92=

2>

8

844

24

844

24ii

.

Skjuin kc whtnr niptenl ef 8 `r 9== 3448>444

2>

8ii

.

\ein thgnf tk nipty t`n rnsnrvker 9

344

=8244`rs 9 >2 `rs.

REA@\ MERM^JHR MUJEFLNR]kjels jegn mermujhr pejjhrs, mermujhr pnfmejs, inhsurefa

ohrs, rkhl rkjjnrs hfl ahs myjeflnrs, ntm., hrn shel tk bn

ef myjeflremhj s`hpn.

Ef iht`nihtemhj tnris,h rea`t mermujhr myjeflnr es h skjel

anfnrhtnl by t`n rnvkjutekf kc h rnmthfajn hbkut ets selns.

Jnt t`n rnmthfajn HBML rnvkjvn hbkut ets seln HB, sk hs tk lnsmrebn h

rea`t mermujhr myjeflnr hs s`kwf ef t`n ceaurn.

Uku iust `hvn kbsnrvnl t`ht t`n mrkss-snmtekf kc h rea`t mermujhr myjeflnr hrn mermjns mkfarunft hfl phrhjjnj tk nhm`kt`nr.

Jhtnrhj ]urchmn

(murvnl surchmn)

Bkttki Nfl

\kp Nfl


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Myjeflnrs Fkt Rea`t Mermujhr\`nrn hrn twk mhsns w`nf t`n myjeflnr es fkt h rea`t mermujhr myjeflnr.Mhsn-E ? Ef t`n ckjjkwefa ceaurn, wn snn h myjeflnr, wèm` es mnrthefjy, but es fkt ht rea`t hfajns tk t`n

bhsn, ]k wn mhffkt shy et es h rea`t mermujhr myjeflnr,

Mhsn-EE ? Ef t`n ckjjkwefa ceaurn, wn snn h myjeflnr, wet` h fkf-mermujhr bhsn hs t`n bhsn es fkt mermujhr.]k wn mhffkt shy et es h rea`t mermujhr myjeflnr,

RNIHRG ? ^fjnss sthtnl kt`nrwesn, `nrn ef tès m`hptnr t`n wkrl myjeflnr wkujl inhf h rea`tmermujhr myjeflnr.\`n ckjjkwefa hrn lncefetekfs kc skin tnris rnjhtnl tk h rea`t mermujhr myjeflnr ?

(e) \`n rhleus kc hfy mermujhr nfl es mhjjnl t`n rhleuskc t`n rea`t mermujhr myjeflnr.\ùs, ef t`n hbkvn ceaurn, HL hs wnjj hs BM es h rhleus kc t`n myjeflnr .

(ee) \`n jefn okefefa t`n mnftrns kc mermujhr nfls kc t`n myjeflnr, es mhjjnl t`n hxes kc t`n rea`t mermujhr myjeflnr.Ef t`n hbkvn ceaurn, t`n jefn HB es t`n hxes kc t`n myjeflnr. Mjnhrjy, t`n hxes es pnrpnflemujhr tk t`n mermujhrnfls.

RNIHRG ? Ec t`n jefn okefefa t`n mnftrns kc mermujhr nfls kc h myjeflnr es fkt pnrpnflemujhr tk t`n mermujhrnfls, t`nf t`n myjeflnr es fkt h rea`t mermujhr myjeflnr.(eee) \`n jnfat` kc t`n hxes kc t`n myjeflnr es mhjjnl t`n `nea`t kr jnfat` kc t`n myjeflnr.(ev) \`n murvnl surchmn okefefa t`n twk bhsns kc h rea`t mermujhr myjeflnr es mhjjnl ets jhtnrhj surchmn.CkriujhnCkr h rea`t mermujhr myjeflnr kc rhleus 9 r ufets & `nea`t 9 ` ufets, wn `hvn ?

Hrnh kc nhm` mermujhr nfl 9 r2sq. ufets.

Murvnl (Jhtnrhj) ]urchmn Hrnh 9 (2r`) sq. ufets.

\kthj ]urchmn Hrnh 9 Murvnl ]urchmn Hrnh+ Hrnh kc twk mermujhr nfls.

9 (2r` + 2r2) sq. ufets.9 W2r (` + r)Y sq. ufets.

Skjuin kc myjeflnr 9 r2` mubem ufets.\`n hbkvn ckriujhn hrn hppjemhbjn tk skjel myjeflnrs kfjy.


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@kjjkw Rea`t Mermujhr Myjeflnrs

]kjels jegn erkf pepns, rubbnr, tubns, ntm., hrn ef t`n s hpn kc `kjjkw myjeflnr.H skjel bkuflnl by twk mkhxehj myjeflnrs kc t`n shin `nea`t hfl leccnrnft rhlee es mhjjnl h `kjjkwmyjeflnr

CkriujhnCkr h `kjjkw myjeflnr kc `nea`t ` hfl wet` nxtnrfhj hfl eftnrfhj rhlee R hfl r rnspnmtevnjy, wn `hvn ?\`emgfnss kc myjeflnr 9 (R r) ufets.Hrnh kc h mrkss-snmtekf 9 (R2 r2) sq. ufets.

9 (R2 r2) sq. ufets.Murvnl (Jhtnrhj) ]urchmn Hrnh 9 (Nxtnrfhj Murvnl ]urchmn Hrnh)

+ (Eftnrfhj Murvnl ]urchmn Hrnh)9 (2R` + 2r`) sq. ufets 9 2` (R + r) sq. ufets.

\kthj ]urchmn Hrnh 9 (Murvnl ]urchmn Hrnh) + 2 (Hrnh kc Bhsn Refa)9 W(2R` + 2r`) + (R2 r2)Y sq. ufets9 2(R` + r` +vR2 r2) sq. ufets

Skjuin kc Ihtnrehj 9 (R2 r2) ` mubem ufetsSkjuin kc @kjjkw rnaekf 9 r2` mubem ufets

Nx. 3 2.2 mu li kc brhss es tk lrhwf eftk h myjeflremhj wern kc lehintnr 4.>4 mi. Cefl t`n jnfat` kc t`n wern.]kj. Skjuin kc brhss 9 2.2 mu li 9 (2.2x84x84x84) mi=9 2244 mi=. Jnt t`n rnquernl jnfat` kc wern bn x mi.

\`nf, ets vkjuin 9

mixmixr 2>.42>.4

1

22)( =2

22442>.42>.41

22 x

imix 882882442>.42>.4

8

22

12244

.

@nfmn, t`n jnfat` kc wern es 882 i.Nx. 1 H wnjj 8< i lehintnr es lua 0 i lnnp. \`n nhrt` thgnf kut kc et hs bnnf nvnfjy sprnhl hjj hrkufl et tk h

welt` kc 28 i tk ckri hf nibhfginft. Cefl t`n `nea`t kc t`n nibhfginft.

]kj. Skjuinkc nhrt` lua kut crki t`n wnjj 9 r2` 9 == 82=2011122 ii

.

Hrnh kc t`n nibhfginft 9 222222 2=8428=>1

22})20{(

1

22)( iiirR

.

@nea`t kc t`n nibhfginft 9 miminibhfginftkcHrnh

kutluanhrt`kcSkjuin=.>=.844

2=84

82=2

.


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Nx.88 \`n tkthj surchmn hrnh kc h rea`t mermujhr mkfn kc sjhft `nea`t 8= mi us 54mi2.

Mhjmujhtn ? (e) ets rhleus ef mi, (ee) ets vkjuin ef mi=, ef tnris kc .

]kj. Aevnf ? sjhft `nea`t, 8= mi.Jnt, rhleus 9 r mi hfl `nea`t 9 ` mi.

(e) \kthj surchmn hrnh 9 r )Y8=(W)r( rr mi2.

4)>)(80(4548=54)8=( 2 rrrrrr

r 9 > WFnajnmtefa r 9 - 80, hs rhleus mhffkt bn fnahtevnY

Rhleuskc t`n mkfn 9 > mi.

(ee) 2222 >8= r`

9 Skjuin kc t`n mkfn=2

82>>=

8

=

8

mi`r

9 844mi=.

Nx.82 H aerj cejjs h myjeflremhj bumgnt =2 mi ef `nea`t hfl 80 mi ef rhleus wet` shfl. ]`n niptens t`n bumgnt kf t`narkufl hfl ihgns h mkfemhj `nhp kc t`n shfl. Ec t`n `nea`t kc t`n mkfemhj `nhp es 2< mi, cefl ?

(e) ets rhleus, (ee) ets sjhft `nea`t.

]kj. @nea`t kc myjeflremhj bumgnt, @ 9 =2 mi.

Rhleus kc myjeflremhj bumgnt, R 9 80 mi.

Skjuin kc shfl 9 R2@

=28080

1

22mi=.

(e) @nea`t kc mkfemhj `nhp, ` 9 2< mi.

Jnt t`n rhleus kc t`n mkfemhj `nhp bn r mi.

\`nf, vkjuin kc mkfemhj `nhp

2 i hfl t`n `nea`t kc t`n myjeflremhj phrt es >4 i. Ec t`n lehintnr kc t`n bhsn es 830 i, cefl t`n quhftetykc mhfvhs rnquernl tk ihgn t`n tnft. Hjjkw 24% nxtrh ckr ckjls hfl stetm`efa.Aevnf ykur hfswnr tk t`n fnhrnst i2.

]kj. Rhleus kc t`n tnft,

2

830r i 9 0< i.

@nea`t kc t`n tnft 9 0> i.@nea`t kc t`n myjeflremhj phrt, @ 9 >4 i.

@nea`t kc t`n mkfemhj phrt, ` 9 (0> >4) i 9 => i.]jhft `nea`t kc t`n mkfemhj phrt, 0208)0( 2222 r` i 9 58i.Xuhftety kc mhfvhs rnquernl 9 Murvnl surchmn hrnh kc t`n tnft

9 Murvnl surchmn hrnh kc t`n myjeflremhj phrt+ Murvnl surchmn hrnh kc t`n mkfemhj phrt9 2r@ + )2( @rr

9 >44


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Nx.8> Crki h skjel myjeflnr kc `nea`t =4 mi hfl rhleus 1 mi, h mkfemhj mhvety kc `nea`t 2< mi hfl kc bhsn rhleus 1mi es lrejjnl kut. Cefl t`n vkjuin hfl t`n tkthj surchmn kc t`n rnihefefa skjel.

]kj. Rhleus, r 9 1 mi.@nea`t kc t`n myjeflnr, @ 9 =4 mi.@nea`t kc t`n mkfn, ` 9 2< mi.

]jhft `nea`t kc t`n mkfn, 2>32>)1()2


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8=

]T@NREMHJ ]@NJJ\`n skjel nfmjksnl bntwnnf twk mkfmnftrem sp`nrns tk mhjjnl h sp`nremhj s`njj.

CkriujhCkr h sp`nremhj s`njj wet` nxtnrfhj rhleus 9 R hfl eftnrfhj rhleus 9 r, wn `hvn ?

\`emgfnss kc s`njj 9 (R r) ufets.Kutnr surchmn hrnh 9 < R2sq. ufets.Effnr surchmn hrnh 9< r2sq. ufets.Skjuin kc ihtnrehj 9


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8 mi hfl t`n `nea`t kc t`n mkfn es 84.> mi. \`n kutnr s`njj HBMLNC ess`hlnl hfl es fkt cejjnl wet` emn-mrnhi. HN 9 LM 9 4.> mi, HBNC hflBMCL. Mhmjmujhtn?(e) t`n vkjuin kc t`n emn-mrnhi ef t`n mkfn (t`n ufs`hlnl pkrhtekf

efmjulefa t`n `niesp`nrn) ef mi=7(ee) t`n vkjuin kc t`n kutnr s`njj (t`n s`hlnl pkrtekf) ef mi=. Aevn

ykur hfswnr tk t`n fnhrnst mi=.]kj. Rhleus kc `niesp`nrn, R9 HA 9 =.> mi.

Nxtnrfhj rhleus kc mkfemhj s`njj, R 9 HA 9 =.> mi.

Eftnrfhj rhleus kc mkfemhj s`njj, r 9 NA 9 (HA HN) 9 (=.> 4.>) mi 9 = miFkw, BA ~ NCA.

5>.=

=

4>.8 CA

CA

HA

NA

BA

CAmi.

]k, eftnrfhj `nea`t kc mkfemhj s`njj, ` 9 CA 9 5 mi.


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8>

(e) Skjuin kc emn-mrnhi9 Skjuin kc `niesp`nrn + Eftnrfhj vkjuin kc mkfemhj s`njj

9 )2(=

8

=

8

=

2 2=2= `rR`rR

9

08

mi rnspnmtevnjy. \`n rhlee kc t`n `niesp`nremhj hflmkfemhj phrt hrn t`n shin hs t`ht kc t`n myjeflremhj phrt. Mhjmujhtn t`n surchmn hrnh kc t`n `nea`t kc t`nmkfemhj phrt es 82 mi.

]kj. \`n tky es ef t`n s`hpn s`kwf bnjkw ?Rhleus kc t`n `niesp`nremhj phrt 9 > mi, Murvnl surchmn hrnh kc t`n @niesp`nremhj phrt

2222 )>4()>(2W2 mimir .

Myjeflremhj phrt `hs rhleus 9 > mi hfl `nea`t 9 8= mi. Murvnl surchmn hrnh kc t`n myjeflremhj phrt 9 22 )8=4()8=>2( mimir` .Mkfemhj phrt `hs rhleus 9 > mi hfl `nea`t 9 82 mi.

Ets sjhft `nea`t .8=8358 22 mi

Murvnl surchmn hrnh kc t`n mkfemhj phrt 9 r

922 )3>()8=>( mimi

@nfmn, t`n surchmn hrnh kc t`n tky 9 (>4+ 8=4 + 3> ) mi29 (2 ) mi

2.

9 22 1141222 mimi

.

Hjsk, vkjuin kc t`n tky 9===22=

=

8>2>844=2>

=

2>4

=

8

=

2mimimi@r`rr


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83

Nx.85 \`n kutnr hfl effnr lehintnrs kc h `niesp`nremhj bkwj hrn 81 mi hfl 8> mi rnspnmtevnjy. Cefl mkstkc pkjes`efa et hjj kvnr ht 2> phesn pnr mi2. (\hgn 9 22/1).

]kj. Kutnr rhleus2

81 mi, Effnr rhleus

2

8> mi.

Hrnh kc kutnr surchmn 9 2 R2 9

22

2

2

205

2

812 mimi

.

Hrnh kc effnr surchmn 9 2 r29

22

2

2

22>

2

8>2 mimi

.

Hrnh kc t`n refa ht t`n tkp 9 (R2 r2) 9 W(0.>)2 - (1.>)2Y mi29 (83 ) mi2.

\kthj hrnh tk bn pkjes`nl 2832

22>

2

205mi

.

222 0>01

2221=)21=( mimimi

.

Mkst kc pkjes`efa t`n bkwj 9 Rs

8442>0>0 9 Rs. 284.

Nx.24 H mkfemhj vnssnj kc rhleus 3 mi hfl `nea`t 0 mi es mkipjntnjy cejjnl wet` whtnr. H sp`nrn es jkwnrnleftk t`n whtnr hfl ets sezn es sum` t`ht w`nf et tkum`ns t`n selns, et es oust eiinrsnl. Q`ht crhmtekf kcwhtnr kvnrcjkws;

]kj. Rhleus kc t`n mkfemhj vnssnj, R 9 HM 9 3 mi.@nea`t kc t`n mkfemhj vnssnj, @ 9 KM 9 0 mi.Jnt t`n rhleus kc t`n sp`nrn bn r.\`nf, TM 9 TL 9 3 mi.Wjnfat`s kc twk thfanfts crki hf nxtnrfhj pkeft tk h mermjn hrn nquhjY

KH 98484430

2222 HMKM

mi.

KL 9 (KH HL) 9 (84 3)KT 9 (KM TM) 9(0 R).Ef rea`t hfajnl KLT, wn `hvn ?

KT29 KL2+ TL2 (0 R)29


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CR^]\Î

CR^]\Î KC H REA@\ MERM^JHR MKFNEf kur lhy-tk-lhy jecn wn mkin hmrkss h fuibnr kc skjels kc t`n s`hpn hs s`kwf ef t`n ceaurn. Ckr nxhipjn, hbumgnt kr h ajhss tuibjnr. Qn kbsnrvn t`ht tès typn kc skjel es h phrt kc h rea`t mermujhr mkfn hfl es kbthefnl w`nft`n mkfn es mut by, h pjhfn phrhjjnj tk t`n bhsn kc t`n mkfn.

Ec rea`t mermujhr mkfn es mut kcc by h pjhfn phrhjjnj tk ets bhsn, t`n pkrtekf kc t`n mkfn bntwnnf t`n pjhfn hflt`n bhsn kc t`n mkfn es mhjjnl h crustui kc t`n mkfn.

Qn mhf snn tès prkmnss crki t`n ceaurns aevnf bnjkw?\`n jkwnr pkrtekf ef ceaurn es t`n crustui kc t`n mkfn. Et `hs twk phrhjjnj cjht mermujhr bhsns, ihrg hs Bhsn (8) hflBhsn (2). H murvnl surchmn okefs t`n twk bhsns.

\`n jefn snainft IF okefefa t`n mnftrns kc t`n twk bhsns es mhjjnl t`n `nea`t kc t`n crustui. Lehintnr ML kcBhsn (2) es phrhjjnj tk lehintnr HB kc bhsn (8). Nhm` kc t`n jefn snainfts HM hfl BL es mhjjnl t`n sjhft `nea`t kct`n crustui. Qn kbsnrvn crki t`n ceaurns (e) hfl (ee) t`ht,

8. @nea`t kc t`n crustui 9 (t`n `nea`t kc t`n mkfn KHB) (t`n `nea`t kc t`n mkfn KML)2. ]jhft `nea`t kc t`n crustui 9 (t`n `nea`t kc t`n mkfn KHB ) (t`n `nea`t kc t`n mkfn)

Skjuin kc h Crustui kc h Rea`t Mermujhr MkfnJnt ` bn t`n `nea`t 7 r8hfl r2bn t`n rhlee kc t`n twk bhsns (r86 r2) kc crustrui kc h rea`t mermujhr mkfn.\`n crustui es ihln crki t`n mkipjntn mkfn KHB by muttefa kcc t`n mkfemhj phrt KML. Jnt `8bn t`n `nea`t kc


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t`n mkfn KHB hfl `2bn t`n `nea`t kc t`n mkfn KML.@nrn, `29 `8 `.]efmn rea`t hfajnl trehfajns KFL hfl KIB hrn seiejhr, t`nrnckrn, wn `hvn?

8

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8 2228

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2

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rrrr`

9 })2()()2(){(=

bhsnkchrnhbhsnkchrnhbhsnkchrnhbhsnkchrnh`

Murbnl ]urchmn Hrnh kc h Crustui kc h Rea`t Mermujhr Mkfn

Jnt ` bn t`n `nea`t, bn t`n sjhft `nea`t hfl r8,r2bn t`n rhlee kc t`n bhsns w`nrn r86 r2.Ef ceaurn (e), wn kbsnrvn NB 9 r8 r2

Hhl2

28

22 )( rr`

2282 )( rr`

Ef ceaurn (ee, wn `hvn KHB hs t`n mkipjntn mkfn crki w`em` mkfn KML es mut kcc tk ihgn t`n crustui HBLM.


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Jnt bn t`n sjhft `nea`t kc t n mkfn KHB hfl 2bn t`n sjhft `nea`t kc t`n mkfn KML.

]efmn, KIB hrn seiejhr,

8

2

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2

r

r

8

2

8

8

r

r

28

88

rr

r

Fkw,

288

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r 28

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rrr

Murvnl surchmn hrnh kc crustui HBML9 (Murvnl surchmn hrnh kc mkfn KHB) (Murvnl surchmn hrnh kc mkfn KML)

9

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\`nrnckrn, murvnl surchmn hrnh kc crustui 9 )( 28 rr .\kthj surchmn Hrnh kc h Crustui kc h skjel Rea`t Mermujhr MkfnJnt ` bn t`n `nea`t, bn t`n sjhft `nea`t hfl r8, r2t`n rhlee kc t`n bhsns w`nrn r86 r2hs s`kwf ef ceaurn.

\kthj surchmn hrnh kc t`es crustui9 Murvnl surchmn hrnh + Hrnh kc Bhsn 8 + Hrnh kc Bhsn 2

92

228 )( rrr

Hrnh kc t`n Inthj ]`nnt ^snl \k Ihgn h BumgntH bumgnt es ef t`n s`hpn kc h crustui kc h rea`t mermujhr `kjjkw mkfn.Jnt ` bn t`n lnpt`, bn t`n sjhft `nea`t, r8bn t`n rhleus kc t`n tkp hfl r2bn t`n rhleus kc t`n bkttki hs s`kwf ef ceaurn

\`n hrnh kc t`n inthj s`nnt usnl ckr ihgefa t`n bumgnt9 Kutnr (kr effnr) murvnl surchmn hrnh + Hrnh kc bkttki

92

228 )( rrr


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Nx.28 H bumgnt kc `nea`t 83 mi hfl ihln up kc inthj s`nnt es ef t`n ckri kc crustui kc h rea`t mermujhr mkfn wetrhlee kc ets jkwnr hfl uppnr nfls hs = mi hfl 8> mi rnspnmtevnjy. Mhjmujhtn?

(e) t`n `nea`t kc t`n mkfn kc wèm` t`n bumgnt es h phrt.

(ee) t`n vkjuin kc whtnr wèm` mhf bn cejjnl ef t`n bumgnt.

(eee) t`n sjhft `nea`t kc t`n bumgnt.

(ev) t`n hrnh kc t`n inthj s`nnt rnquernl tk ihgn t`n bumgnt.

]kj. Jnt HBML bn t n bumgnt wèm` es crustui kc h mkfn wet` vnrtnx 4 (hs s`kwf ef ceaurn). Jnt KF 9 x miKIMKHB

8>

=

83

x

x

IM

FB

KI

KF

>

8

83

x

x 883> xx

83< x x 9 W

=

8mi

=Y82241>W mi

=8 mi

mimimi 2>32>)24()8>( 22 ]jhft `nea`t kc mkfn kc rhleus = mi

mimi >)=()2>(

\`n hrnh kc t`n inthj s`nnt 2)( rrR 222 5=34)=()=8>(24 mimi

2=35 mi Fktn. \`n hrnh kc t`n inthj s`nnt usnl 9 M.]. kc jhranr mkfn M.]. kc sihjjnr mkfn + Hrnh kc t`n bhsn kc t`n bumgn

222 Y58>=1>WY)=(=>8>2>W mimi 2=35 mi


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Nx.22 H bumgnt es ef t`n ckri kc h crustui kc h mkfn, lnpt` es 8> mi hfl t`n lehintnrs kc t`n tkp hfl t`n bkttkihrn >3 mi hfl miMhphmety kc t`n bumgnt }{

=

8 22rRrR`

=22 )}28)(20()28()20{(8>22=

8mi

=}>005>22 mi=

9 20


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Nx.2< \`n `nea`t kc h mkfn es


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2=

HKB hfl MKL hrn lehintnrs kc t`n jkwnr hfl uppnr chmns kc t`n crustui. Lrhw LN KB. Jnt K T 9 xEf rea`t hfajnl LNB,

LB29 BN2+ LN2

)'(22 `KKLN`BN

22225

Documents

13. Surface Area and Volume