Математика за 2. разред средње школе - 22179 by Zavod za udžbenike

ГРАДИМИР ВОЈВОДИЋ

РАДИВОЈЕ ДЕСПОТОВИЋ

ВОЈИСЛАВ ПЕТРОВИЋ

РАTKO TOШИЋ

БРАНИМИР ШЕШЕЉА

MATEMATIKA

za II razred sredwe {kole

др Мирко Јанц

др Илија Ковачевић

Божидар Манић

Владимир Јечмен

Вера Николић

Душан Богдановић

Татјана Костић

Милорад Марјановић

в. д. директора

37.016:51(075.3)

МАТЕМАТИКА : за II разред средње школе :

програми са 3 часа наставе математике недељно / Градимир Војводић ... [и др.]. - 17. изд. - Београд : Завод за уџбенике, 2023 (Београд : Службени гласник). - 279 стр. : граф. прикази ; 24 cm

Тираж 1.500.

ISBN 978-86-17-20952-8

1. Војводић, Градимир, 1947- [autor]

COBISS.SR-ID 122578953

ISBN: 978-86-17-20952-8

1992-2023.

M6 M7 M8 M9 M10 Ҳ M11

ϐүҵҸҬҲҼүҴһҼҪӂҼҪҶҹҪҷҲһҲҼҷҲӑҲҶһҵҸҭҸҶ ҹүҼҲҼҸҶ ҷҪҶүӓүҷҲһҽұҪҺҪҮ һҪұҪҲҷҼүҺүһҸҬҪҲҲҶҽӁүҷҲӀҲҶҪ

ϞҪҴҬҲҶһҵҸҭҸҶӂҼҪҶҹҪҷүһҽҲҹҸӑүҮҲҷүҲһҼҸҺҲӑһҴүҷҪҹҸҶүҷү

үҴҲһҵҽӁҪӑүҬҲһҲһҼүҶҪҮҬүҴҬҪҮҺҪҼҷүӑүҮҷҪӁҲҷүһҪҮҬүҷүҹҸұҷҪҼү

PҬҺӂҲҷBұBҺҽҫӒFҷFҹҲҺBҶҲҮF

ϓBҹҺFҶҲҷBҹPҵҲFҮBҺB ϓBҹҺFҶҲҷBҴҬBҮҺB ҹҺBҬPҽҭҵPҭҹBҺBҵFҵPҹҲҹFҮB ϓBҹҺFҶҲҷBҴPӀҴF ϖBҬBҵҲFҺҲKFҬҹҺҲҷӀҲҹϓBҹҺFҶҲҷBҹҺҲұҶF ϓBҹҺFҶҲҷBҹҲҺBҶҲҮF ϓBҹҺFҶҲҷBұBҺҽҫӒFҷFҹҲҺBҶҲҮF

ҼүҹүҷһҪҺүҪҵҷҲҶүҴһҹҸҷүҷҼҸҶ ϑҴһҹҸҷүҷӀҲӑҪҵҷҪҾҽҷҴӀҲӑҪ

ϔҷҬүҺұҷүҾҽҷҴӀҲӑү

0ύϜ5 "5& "

0ҫҺҼҷBҹPҬҺӂ ώBӒBҴ ҴҽҹBҲұBҺҽҫӒFҷBҴҽҹB

PҬҺӂҲҷFҬBӒҴB ҴҽҹFҲұBҺҽҫӒFҷFҴҽҹF

PҬҺӂҲҷBҹҺBҬPҭҬBӒҴB

PҬҺӂҲҷBҹҺBҬFҴҽҹF

PҬҺӂҲҷBҹҺBҬFұBҺҽҫӒFҷFҴҽҹF ϓBҹҺүҶҲҷBҬBӒҴB ҴҽҹFҲұBҺҽҫӒFҷFҴҽҹF ϓBҹҺүҶҲҷBҬBӒҴB ϓBҹҺүҶҲҷBҴҽҹF ϓBҹҺүҶҲҷBұBҺҽҫӒFҷFҴҽҹF ҾFҺBҲҵPҹҼB

υүҮҷҪӁҲҷҪDPT

TJO

= |AB| |BC | ҲҸҷҪһүҷҪұҲҬҪ

1 DPT α = |AB | |AC |

a Ҳ b ҲҼҸ

B a =5 b =12ҫ a =3 b =5, 2

B a =8 c =10ҫ a =9, 8 c =12, 6

TJO 25◦ TJO 30◦

DPT 25◦ DPT 30◦ DPT

ϚһҷҸҬҷҲҼҺҲҭҸҷҸҶүҼҺҲӑһҴҲҲҮүҷҼҲҼүҼҲ

ϓόϐόҠϔ

ϚҮҺүҮҲҸһҼҪҵүҼҺҲҭҸҷҸҶүҼҺҲӑһҴүҾҽҷҴӀҲӑүҪҴҸӑүҮҪҼҸ

B TJO α = 5 13 ҫ DPT α =0, 25

ϚҮҺүҮҲҸһҼҪҵүҼҺҲҭҸҷҸҶүҼҺҲӑһҴүҾҽҷҴӀҲӑүҪҴҸӑүҮҪҼҸ

B UH α = 4 5 ҫ DUH α =0, 41

ϐҸҴҪҰҲҮҪӑү

B TJO2 α DUH2 α + DPT 2 α UH2 α =1

ҫ TJO2 α(1+ DUH2 α)+ DPT 2 α(1+ UH2 α)=2

ϐҸҴҪҰҲҮҪӑү TJO α + DPT β

ҝҹҺҸһҼҲҲұҺҪұү B 1

ϐҸҴҪҰҲҮҪӑүұҪһҬүҸӂҼҺүҽҭҵҸҬү

α ҺҪұҵҲӁҲҼүҸҮ 45◦ DPT α 1 UH α TJO α

ҝӂҴPҵBҶBһҽһFҷBKӁFӂӔFҽҹPҼҺFҫӒBҬBҵFҼұҬҹFҼPӀҲҾҺFҷFҶBҼFҶBҼҲӁ ҴFҼBҫҵҲӀFҴPKFһҽҲҶBҵFҷBһҵPҬ

ҶҽҴPҼҺҹҷPһҼҬBҺBҷF ҮPҹҽӓBҬBҷFҲҽһBҬҺӂBҬBҷFҮFӀFҷҲKBҶB ҲҼPҷBKҹҺFҫFұ ҽҹPҼҺFҫFҺBӁҽҷBҺBϖBҴPһFҼPҹPҹҽҵBҺҷPҴBҰF ҺBӁҽҷBҵPһF ҹFӂҴFi

TJO(90◦ α)= DPT α, UH(90◦ α)= DUH α, DPT(90◦ α)= TJO α, DUH(90◦ α)= UH α,

TJO

PҶPӔҽҴBҵҴҽҵBҼPҺB ҮҲҭҲҼҺPҷB ҮPҫҲKBҶPҮҲҺFҴҼҷP TJO 20◦ 48 = TJO 20+ 48 60 = TJO 20, 8◦ =0, 355106962. ҬҲҮBҷBӂӓҲҹFҺһPҷBҵҷҲҺBӁҽҷBҺҲҲҶBKҽҴBҵҴҽҵBҼPҺFһBҺBұҷPҬҺһҷҲҶ

TJO 20, 8◦ =0, 35510696240813705136617948548259,

DPT 20◦ 50 =0, 93462 p ұҪ 3 =31

DPT 20◦ 53 =0, 93431

ҽ ҴBҵҴҽҵBҼPҺBDPT 20◦ 53 =0, 934308205

DUH 20◦ 10 =2, 7228 p ұҪ 9 =219

DUH 20◦ 19 =2, 7009

ҽҷBҶPUH 20◦ 19 = 0, 370241959 BұBҼҲҶ

DUH 20◦ 19 =1/ UH 20◦ 19 =2, 700936439.

ҭPҷPҶFҼҺҲKһҴFҾҽҷҴӀҲKF ҹPұҷBҼ ҷҲKFKFҮҷPһҼBҬҷPϖBҵҴҽҵBҼPҺPҶһFҼPKFҮҷPһҼBҬҷP ҷBҵBұҲҹҺҲҶFҷPҶҾҽҷҴӀҲKFJOWTJO

ϞBҫҵҲӀBҹҺҲҺPҮҷҲҿҬҺFҮҷPһҼҲһҲҷҽһB ҴPһҲҷҽһBҲҼBҷҭFҷһBһBҼBӁҷҲҿ ҮFӀҲҶBҵBҮPҫҲKFҷҪӑүҹPҶPӔҽҹҺPҭҺBҶB."5-"#ҲҮҪҼҪӑүҷҪһҵүҮүӔҸӑһҼҺҪҷҲ

= TJO 1 0, 74281=47, 97133779

α =47◦ +60 0, 97133779=47◦ +58, 2802674 =47◦58 +60 · 0, 2802674 ≈ 47◦5817 . ҺҲҶүҺ ϚҮҺүҮҲҶҸ ҽҭҪҸ α ҪҴҸӑүDPT α =0, 23849

= DPT 1 0, 23849=76, 20256393◦ ҲұҵҪұҲ

B TJO 19◦38ҫ TJO 64◦17 Ҭ DPT 27◦43ҭ DPT 72◦23

l =0, 993563 0, 002536 DPT 2ϕ g =9, 806059 0, 025028 DPT ϕ

b = c DPT α =93 DPT 42◦ , 3833=93 0, 73865 ≈ 69 b = 932 632 ≈ 69 DN

a =19 α =21◦ 17

a =28, 602 β =63◦ 27

c =40 α =71◦ 36

c =108, 96 β =57◦ 6

c =32 a =16, 5

c =13, 211 b =7, 821

a =78 b =42

a =6, 35 b =9, 72

ϚҮҺүҮҲҮҽҰҲҷҽһүҷҴүҮҺҬүҼҪӁҲӑҪӑүҬҲһҲҷҪN ҪҴҸһҽҷӁүҬҲұҺҪӀҲҹҪҮҪӑҽҹҸҮ

|AB| = d · UH α |

ҝҹҺҪҬҸҽҭҪҸҷҲҴҽӑүҮҪҼҪһҼҺҪҷҲӀҪ

ϐҲӑҪҭҸҷҪҵүҺҸҶҫҪһҽDNҲDNϔұҺҪӁҽҷҪӑһҼҺҪҷҲӀҽҲҽҭҵҸҬүҺҸҶҫҪ

ϖҸҵҲҴҲӑүүҵүҬҪӀҲҸҷҲҽҭҪҸϝҽҷӀҪҪҴҸӂҼҪҹҮҽҰҲҷү NҫҪӀҪһүҷҴҽҮҽҭҽ N

ϒүӒүұҷҲӁҴҪҹҺҽҭҪҷҪӑүҮҷҸҶҮҲӑүҵҽҼүҺүҷҪҲҶҪҽһҹҸҷ

B ϖҸҵҲҴҲӑүҷҪҭҲҫҷҲҽҭҪҸ

C ҪҴҸӑҸӑӑүҬҲһҲҷҲҷҪҮҿҲҹҸҼүҷҽұҸҶҼҪӁҴҪҴҸӑҪӑүҸҮҹҸӁүҼҴҪҽҮҪӒүҷҪN

F1 Ҳ F2 ҴҸӑүұҪҴҵҪҹҪӑҽҽҭҪҸҸҮ

(am)n = am·n m,n ∈ N

Ҫ 0, 53ڈ 2, 8012Ҭ 0, 00008

Ϝүӂүӓү Ҫ 0, 53= 5 10 + 3 102 = 53 102 ڈ 2, 8012=2+ 8012 104 Ҭ 0, 00008= 8 105

224 =23 8 =(23 )8 =88

316 =32 8 =(32 )8 =98

Ү a 3 b 1 Ӌ 7xz 4 F 6(a + b) 5 Ұ 12a 1 (b c) 5

ҝһҬҪҴҸҶҹҺҲҶүҺҽҷҪұҷҪӁҲҸҭҺҪҷҲӁүӓүұҪҹҺҸҶүҷӒҲҬү

0, 0017

ҫ ҶҪһҪϓүҶӒүӑүҮҷҪҴҪӑүU

Ҭ ӁҸҬүҴҸҬҸҺҭҪҷҲұҪҶҲҶҪҬҲӂүҸҮӔүҵҲӑҪ ҭ ҶҪһҪҪҼҸҶҪҬҸҮҸҷҲҴҪӑүҮҷҪҴҪӑү H

Ү ҺҪһҼҸӑҪӓүҸҮLNһҬүҼҵҸһҼҽҬҪҴҽҽҶҽҹҺүӋүұҪ

ҮҪҭҺҪҾҲҴҽҾҽҷҴӀҲӑүҹҺҲҹҪҮҪҴҸҸҺҮҲҷҪҼҷҲҹҸӁүҼҪҴ ҸҮҷҸһҷҸҮҪӑүҫҺҸӑ ҷҽҵҪҾҽҷҴӀҲӑү

ҮҪҭҺҪҾҲҴҹҺҲҹҪҮҪҹҺҬҸҶҲҮҺҽҭҸҶҴҬҪҮҺҪҷҼҽ ҮҪӑүҭҺҪҾҲҴҾҽҷҴӀҲӑүһҲҶүҼҺҲӁҪҷҽҸҮҷҸһҽҷҪ

f (x)= x 14

Ҫ f (3) Ҳ f ( 3)ҫ f (5) Ҳ f (0)Ҭ f ( 8) Ҳ f (0)

ҭ f ( 6) Ҳ f (8)Ү f ( 3) Ҳ f ( 1)Ӌ f (4) Ҳ f (7)

ϐҪҼҪӑүҾҽҷҴӀҲӑҪ h(x)= x 27 үҺҪӁҽҷҪӑҽӔҲ ҽҹҸҺүҮҲ

Ҫ h(4) Ҳ h( 4)ҫ h( 10) Ҳ h(0)Ҭ h(12) Ҳ h(0)

ҭ h( 28) Ҳ h(45)Ү h( 5) Ҳ h( 2)Ӌ h(7) Ҳ h(9)

ϛҺҸҬүҺҲҮҪҵҲһҵүҮүӔүҼҪӁҴүҹҺҲҹҪҮҪӑҽҭҺҪҾҲҴҽҾҽҷҴӀҲӑү y = x 8

A(2;256) B( 2;256) C( 3; 6561)

ϛҺҸҬүҺҲҮҪҵҲһҵүҮүӔүҼҪӁҴүҹҺҲҹҪҮҪӑҽҭҺҪҾҲҴҽҾҽҷҴӀҲӑү y = x 9

A(2;512) B( 2; 512) C( 3;19683)

ϖҸҺҲһҼүӔҲһҬҸӑһҼҬҪҭҺҪҾҲҴҪҾҽҷҴӀҲӑү

f ( a) Ҳ g( b)

x 35 ϛҸұҷҪҼҸӑүҮҪӑү f (a)=80 g(b)=121

n ҪҴҸӑүҹҸұҷҪҼҸҮҪҭҺҪҾҲҴҾҽҷҴӀҲӑү

A(3;27)ҫ B(4, 5;20, 25)Ҭ C( 2;16)ҭ D( 3; 243)

y 3 =125

y = xn (n 4) ҪҽҹҺҬҸҶҺҪұҺүҮҽҭҺҪҾҲҴҾҽҷҴӀҲӑү y = a ҺҲҶүҷҸҶҼҲҿҾҽҷҴӀҲӑҪ ҽҼҬҺҮҲӔүҶҸҴҸҵҲҴҸҲҴҪҴҬҲҿҴҸҺүҷҪ ҺүӂүӓҪ ҲҶҪӑүҮҷҪӁҲҷҪ

y = x 4

nҼҲҴҸҺүҷҴҪҮҪӑүҺҪҮҲҴҪҷҮҷүҭҪҼҲҬҪҷ ҸҮҷҸһҷҸҪҴҸӑү

(a + bi = c + di) ⇔ (a = c ∧ b = d)

(a + bi)+(c + di)=(a + c)+(b + d)i

(a + bi)(c + di)=(ac bd)+(ad + bc)i

z = a + bi

(z)= a Ҳ*N(

(a + bi)(c + di)=(a + bi)c +(a + bi)di = ac + bci + adi + bdi2 = ac +( 1)bd +(bc + ad)i =(ac bd)+(bc + ad)i.

=64( 1)= 64

x 2 +1=0

B (1, 1)ڈ (1, 0)Ҭ (0, 1)ҭ (0, 0)

Ϝүӂүӓү B (1, 1)=1+ iڈ (1, 0)=1+0i =1

Ҭ (0, 1)=0+1i = iҭ (0, 0)=0+0i =0

ҸҴҪҰҲҶҸҮҪҽҬүҮүҷүҸҹүҺҪӀҲӑүһҴҸҶҹҵүҴһҷҲҶҫҺҸӑүҬҲҶҪҲҶҪӑҽҲһҼҪ ҸһҷҸҬҷҪһҬҸӑһҼҬҪҴҸӑҪҲҶҪӑҽҼүҸҹүҺҪӀҲӑүҽҹҸӒҽҺүҪҵҷҲҿҫҺҸӑүҬҪ

ӑҪ (0, 0)

Ϝүӂүӓү (a,b)+(0, 0)= a + bi +0+0i =(a +0)+(b +0)

1 6 + 1 12

Ҫ ( 4+ i) (16 2i)+(29 i)ҫ 17i +( 11 8i) 14i +6

Ҭ 25=6i 3 5i +13i 49ҭ 52 ((61+40i) ( 10+ i))

Ү (8+ i

Ҫ 6i(8+9i)ҫ 11i( 1+10i)Ҭ ( 2+ i)(3 2i)

ҭ ( 4 i)(5+3i)Ү (9+ √ 12)(7 √ 3)Ӌ ( 2+7i)2

Ҫ 7+5iҫ 6 3iҬ 5 2

8Ӌ (3, 1)ү ( 5, 6)Ұ (0, 2)

ϜҪһҼҪҬҲҮҪҼүұҫҲҺҸҬүҷҪҹҺҸҲұҬҸҮҴҸҶҹҵүҴһҷҲҿӁҲҷҲҵҪӀҪ

Ҫ x 2 +16ҫ 4x 2 +9Ҭ 1+49y 2 ҭ a 2 +3

Ҫ ( 12+16i):(8 4i)=( 2+ i)ҫ ( 23+41i):(1+3i)=(10+11i)

ϔұҺҪӁҽҷҪӑ

(x 2 + x +1)(x 1)= x 3 + x 2 + x x 2 x 1= x 3 1

P2 (α)= aα 2 + bα + c

P2 (x)=2x 2 5x 3 ұҪ x =1

P2 (α)=0ϞҪҴҸӑүҫҺҸӑҴҸҺүҷұҪҼҺҲҷҸҶ

ax 2 + bx +

ax 2 + bx + c =(e1 x + d1 )(x α)+ r1 .

(ex + d)(x α)+ r =(e1 x + d1 )(x α)+ r1 ,

(x α)((ex + d) (e1 x + d1 ))= r1 r. όҴҸҫҲҫҲҵҸ ex + d = e1 x + d1

ax 2 + bx + c =(ax +

x 2 +3x 6=(ex + d)(x 2)+ r.

ϔұӑүҮҷҪӁҪҬҪӓүҶҴҸүҾҲӀҲӑүҷҪҼҪҽұҲһҼүһҼүҹүҷүҷүҹҸұҷҪҼү x ҮҸҫҲӑҪҶҸ 1= e, 3= 2e + d, 6= r 2d, ҸҮҷҸһҷҸ e =1,d =5,r =4. ϖҸҷҪӁҷҸҲҶҪҶҸ x 2 +3x 6=(x +5)(x 2)+4.

ҸҵҲҷҸҶ x 2 +3x 10 ҮүӒҲҬӑүһҪ x 2 ϖҸҵҲӁҷҲҴӑү x +5 ҪҸһҼҪҼҪҴ ҼӑҬҪҰҲ x 2 +3x 6=(x +5)(x 2)

ax 2 + bx + c =(ax + d)(x α)+ r.

ϐҸҸһҼҪҼҴҪ r ҶҸҰүҶҸҮҸӔҲҲҷүҬҺӂүӔҲҮүӒүӓүϖҪҴҸҾҸҺҶҽҵҪ ҬҪҰҲұҪ һҬү x ҼҸҸҷҪҬҪҰҲҲұҪ x = α

ϐҪҴҵү ӑүҮҪҷҴҸҺүҷҼҺҲҷҸҶҪ x 2 3x +2

x1 Ҳ x2

(x3 x1 )(x3 x2 )=0

x3 = x1 ҲҵҲ x3 = x2

Ҫ 2x 2 6x +(1 2i) Ҳ x 2 +(i +1)x +2i ҫ 7x 2 +6x +8

Ҫ ҸҮүҵҲ x 2 8x +12 һҪ x 6 ҲҷҪӋҲҸһҼҪҼҪҴҮүӒүӓҪ ҫ ҸҮүҵҲ 2x 2 5x +8 һҪ x 3

(3a + b)x 2 +(a 3b)x +2

x 2 +2x +2

x 2 5x +6

x 2 8x +7

3x 2 +4x 7

Ҫ x 2 10x 200 һҪ x 20ҫ 2x 2 6x +4

Ҭ x 2 12x +4 һҪ x 2 3

x 2

x 2 5x +6

x 2 5x +6=(x 2)(x 3). όҴҸҹҸҶҷҸҰҲҶҸ

( 5= (2+3), 6=2 · 3).

x(3x +2)=0

3x 2 +12=0

ҴҸҺүҷҲ x1 =0 Ҳ x2 = 2 3

anxn + an 1 xn 1 + + a1 x + a0 =0,an,an 1 ,...,a0 ∈ C,n> 0

ϜүӂҲӑүҮҷҪӁҲҷү Ҫ x 2 =4 3xҫ 8(2 5x)=25x 2 Ҭ x(x 6)=13

ҭ x(2 3x)= x 2 +7x 4

Ү (x 1)(x 2)=3

Ӌ (x 2)(x 3)= x

ү 3x +2 3 = x 7 2x +1

Ұ 2x 4 3x +6 =100x

ϜүӂҲӑүҮҷҪӁҲҷү

Ҫ (5x +2)(3x +1) (4x 5)(4x +5)=37

ҫ x 2 2√3x +1=0 Ҭ x 2 6ix 5=0

ҭ x 2 + x +1=0Ү 2x 2 +3x +4=0

ϜүӂҲӑүҮҷҪӁҲҷү

Ҫ (a 2 b2 )x 2 2a 2 bx + a 2

3 10ҫ 5 6 Ҭ 2+3

5x 2 2x 3=0

x 2 2nx 3n 2 =0

ҮҬүҷүҹҸұҷҪҼүϞҸӑүһҲһҼүҶҸҫҵҲҴҪ

y 2 4y 3=0,

ҷҪһҵүҮүӔҲҷҪӁҲҷόҴҸҸұҷҪӁҲҶҸ

B x =8 y, x · y =15;

=10, x + y =3; ҭ 2(x 3)=6(x y), x · y =4; Ү x 2 +2y 2 =3, x y 1=0.

ϜүӂҲһҲһҼүҶ

B x 2 + y 2 =4a 2 , x + y =2a; ҫ x 2 y 2 = a 2 , x y = a;

Ҭ 2x 2 5xy +3y 2 =48, 3x y =11; ҭ x 2 + y 2 14x 4y =0, 3x 2y 12=0

ϜүӂҲӑүҮҷҪӁҲҷү

Ҫ x 4 13x 2 +36=0ҫ x 4 17x 2 +16=0

Ҭ x 4 17x 2 +12=0 ҭ x 4 +32x 2 369=0

ϜүӂҲһҲһҼүҶӑүҮҷҪӁҲҷҪ

B x 2 + y 2 =25, x 2 +2y 2 =41; ҫ x 2 +4y 2 =20, 3x 2 y 2 =47; Ҭ 1 x2 + 4 y2 =40, 3 x2 5 y2 = 33

1: x = x :(1 x

x 2 + x 1=0

ҽҶҽҸҮҮҲҷҪҺҪҼҺүҫҪҺҪұҮүҵҲҼҲҲұҶүӋҽҬҲӂүҸһҸҫҪϖҪҮҪҫҲҫҲҵҸҸһҸҫҪ

ϖҪҮҪҫҲҫҲӀҲҴҵҲһҼҪҬҸұҲLNIҫҺҰү ҹҽҼҸҮLNҹҺүӂҪҸҫҲһҪҼҪҺҪҷҲӑү ϖҸҵҲҴҪӑүҫҺұҲҷҪӓүҭҸҬҸҭҴҺүҼҪӓҪ

υүҮҷҪҴҪҼүҼҪҹҺҪҸҬҸҽҭҵҸҭҼҺҸҽҭҵҪҬүӔҪӑүҸҮҮҺҽҭүұҪDN ҮҸҴҹҸҬҺӂҲҷҪҼҺҸҽҭҵҪ ҲұҷҸһҲDN2 ϚҮҺүҮҲӓүҭҸҬҸҫҲҶ

ϖҸӑҲҹҺҪҬҲҵҷҲҶҷҸҭҸҽҭҪҸҲҶҪҮҲӑҪҭҸҷҪҵҪ

ϐҲӑҪҭҸҷҪҵҪҹҺҪҬҸҽҭҪҸҷҲҴҪҲұҷҸһҲDNόҴҸһүҮҽҰҲҷҪҹҺҪҬҸҽҭҪҸҷҲҴҪҹҸҬүӔҪ ұҪDNҪӂҲҺҲҷҪұҪDN ҮҲӑҪҭҸҷҪҵҪӔүһүҹҸҬүӔҪҼҲұҪDNϚҮҺүҮҲһҼҺҪҷҲӀү

b2 4ac< 0

ҼҪӁҪҴҪһҪ xҸһҸҶ

= x 2 8x +7

(1, 0) Ҳ (7, 0)

ҬҺүҮҷҸһҼҲҹҺҸҶүҷӒҲҬү x ҶҪӓүҸҮҬҪҰҲҪҴҸ

f (v) >f (u)

f (x)= x 2 8x +7 ҸҹҪҮҪұҪ

f (5) f (6) f (6) Ҳ f (7) ҝҸҬҸҶһҵҽӁҪӑҽҴҪҰүҶҸҮҪҾҽҷҴӀҲӑҪ f (x)= x 2 8x +7 ҺҪһҼүұҪ x> 4 ҼӑұҪһҬүҺүҪҵҷүҫҺҸӑүҬү

ұҪһҬҪҴҸ u v ∈ (s,t) ҬҪҰҲҲұ u<v һҵүҮҲ f (u) <f (v)ϖҪҰүҶҸҮҪҾҽҷҴӀҲӑҪ

ҷҪҸҼҬҸҺүҷҸҶҲҷҼүҺҬҪҵҽ (s,t)

һҵүҮҲ f (u) >f (v)

u v ∈ (s,t) ҬҪҰҲҲұ u<v

v u> 0 ұҪӂҼҸ Ҫ v + u 8 <

f (v) f (u) < 0 Ҽӑ f (v) <f (u) ϐҪҴҵү ҷҪҮҲҷҼүҺҬҪҵҸҶ (−∞, 4) ҾҽҷҴӀҲӑҪ y = x 2 8x +7 ҸҹҪҮҪ ϖҬҪҮҺҪҼҷҪҾҽҷҴӀҲӑҪ

f (x)= x 2 8x +7

f (1)= f (7),f (2)= f (6),f (3)= f (5)

(4 t,f (4 t))

Ҳ (4+ t,f (4+ t))ϞүҼҪӁҴүһҽһҲҶүҼҺҲӁҷүӑүҮҷҪһҪҮҺҽҭҸҶҽҸҮҷҸһҽҷҪҹҺҪҬҽ s ҴҸӑҪҹҺҸҵҪұҲҴҺҸұҼүҶүҴҬҪҮҺҪҼҷүҾҽҷҴӀҲӑүҲҹҪҺҪҵүҵҷҪӑүҸһҲ

(D> 0)

ұҷҪҴҪҽҶҷҸҰҪҴҪ (x 1) Ҳ (x 7)όҴҸӑү x< 1 ҸҷҮҪӑү x< 7 ҹҪӑүҹҺҸҲұҬҸҮ (x 1)(x 7) > 0

x1 <x2

a (x x1 ) Ҳ (x x2 )όҴҸӑү x<x1 ҸҷҮҪӑүҲ x<x2 ҹҪӑүҹҺҸҲұҬҸҮ (x x1 )(x x2 ) ҹҸұҲҼҲҬҪҷ ϓҪӂҼҸ ϐҪҵҲһҽ x x1 Ҳ x x2

ϔһҼҸҼҪҴҸ ҪҴҸӑү x>x2 Ҹҷ

ҮҪӑүҲ x>x1 ҹҪҬҪҰҲ x x1 >

0 Ҳ x x2 > 0ϐҪҴҵүҲҶҪҶҸ (x x1 )(x x2 ) > 0

ϘүӋҽҼҲҶ ҪҴҸӑү x1 <x< x2 ҸҷҮҪӑү x x1 > 0 Ҫ x x2 <

0 ҹҪӑү (x x1 )(x x2 ) < 0

όҴҸӑү D> 0 ұҷҪҴҴҬҪҮҺҪҼ

ҷүҾҽҷҴӀҲӑү y = ax 2 + bx + c

ҲҶҪұҷҪҴҴҸүҾҲӀҲӑүҷҼҪ a ҸһҲҶ ұҪҸҷүҬҺүҮҷҸһҼҲ

x1 Ҳ x2

ҺҲҶүҼҲҶҸҮҪұҪ x ∈ R ҬҪҰҲ f (x)= x 2 = g

+3,α 2 )

g(x)

3)2 һҵ

(α,α 2 )ϞҪҼҪӁҴҪҹҺҲҹҪҮҪҭҺҪҾҲҴҽҾҽҷҴӀҲӑү

(α +3,α 2 )

ҭ y =(x 1)(x 2)Ү y =(x 2)(x 3) (x 3)(4 x)

ϚҮҺүҮҲҷҽҵү үҴһҼҺүҶҷүҬҺүҮҷҸһҼҲҲҴҸҸҺҮҲҷҪҼүҼүҶүҷҪһҵүҮүӔҲҿҴҬҪҮҺҪҼҷҲҿ

y = x 2 + x +1Ү y =2(x 3)(x 1)+(2 3x)(x 4)

ϐҪҼҪӑүҴҬҪҮҺҪҼҷҪҾҽҷҴӀҲӑҪ y =(m 2)x 2 +(1 3m)x +5m 3 m =2ϚҮҺүҮҲ m ҼҪҴҸҮҪҼҪӁҴҪ A(2, 3) ҹҺҲҹҪҮҪҭҺҪҾҲҴҽ ұҪҼҲҶҷҪӋҲүҴһҼҺүҶҼүҾҽҷҴӀҲӑү ϐүҼҪӒҷҸҲһҹҲҼҪӑҴҬҪҮҺҪҼҷүҾҽҷҴӀҲӑү ҷҪӋҲҷҽҵү ҹҺүһүҴһҪ OyҸһҸҶ үҴһҼҺүҶҷү ҬҺүҮҷҸһҼҲ ҲҷҼүҺҬҪҵүҺҪһҼҪ ҸҮҷҸһҷҸҸҹҪҮҪӓҪҲұҷҪҴ

D =( 5)2 4 · 4=9 ҼҸӔүҴҬҪҮҺҪҼҷҪҾҽҷҴӀҲӑҪ

m1 =1 m2 =4

m ∈ (−∞, 1) ∪ (4, ∞)

m 2 5m +4

m ∈ (−∞, 1) ∪ (4, ∞)\{0}

m 2 5m +4

ϏҺҪҾҲӁҴҸҺүӂҪҬҪӓүһҲһҼүҶҪ y = ax 2 + bx + c Ҳ y = mx + n

ϖҸҵҲҴҽҬҺүҮҷҸһҼҼҺүҫҪҮҪҲҶҪ

Ҫ x 2 4x + m> 15ҫ x 2 4x + m< 15

ҝӑүҮҷҪӁҲҷҲ (m +1)x 2 +(m +4)x +2m 1=0, (m = 1)

ҸҮҺүҮҲ m ҼҪҴҸҮҪҺүӂүӓҪӑүҮҷҪӁҲҷүҫҽҮҽ

Ҫ ҺүҪҵҷҪҲҺҪұҵҲӁҲҼҪҫ ҺүҪҵҷҪҲӑүҮҷҪҴҪ

ϜόҞϔҡϖϚ ϜϑҢόώόχϑ

ϚҲҺҪӀҲҸҷҪҵҷҲҶӑүҮҷҪӁҲҷҪҶҪ

ϜүӂҲһҲһҼүҶү

B 3x 2 4y =36, 2x +3y =17; ҫ x 2 1= y, x +1= y.

ϜүӂҲһҲһҼүҶү

B x 2 =8y, 4y x 12=0; ҫ (x 1)(x 2)=3(2y 1), 7y 4(x +1)=0.

ϚҮҺүҮҲҭҺҪҾҲӁҴҲҺүҪҵҷүҴҸҺүҷүӑүҮҷҪӁҲҷү ҽҴҸҵҲҴҸҲҿҲҶҪ 1 4 x 2 x +1=0

ϚҮҺүҮҲҭҺҪҾҲӁҴҲҺүҪҵҷүҴҸҺүҷүӑүҮҷҪӁҲҷү ҽҴҸҵҲҴҸҲҿҲҶҪ 4x 2 12x +7=0

ϚҮҺүҮҲҭҺҪҾҲӁҴҲҺүҪҵҷүҴҸҺүҷүӑүҮҷҪӁҲҷү ҽҴҸҵҲҴҸҲҿҲҶҪ 12x 2 +17x 16=0

√x +6= x ⇔ x +6= x 2 ∧ x 0

⇔ x 2 x 6=0 ∧ x 0

⇔ (x =3 ∨ x = 2) ∧ x 0

x 2 x 6=0)

⇔ (x =3 ∧ x 0) ∨ (x = 2 ∧ x 0) ҴҸҺҲһҼүӔҲҼҪҽҼҸҵҸҭҲӑҽ p ∧ (q ∨ r) ⇔ (p ∧ q) ∨ (p ∧ r))

⇔ (x =3 ∧ x 0) ӑүҺӑү x = 2 ∧ x 0 ⇔⊥, p ∨⊥⇔ p)

⇔ x =3 ӑүҺӑү x =3 ⇒ x 0)

ҼBӁҷPKFҮҷBҹҺBҬBҹBҺBҵFҵҷBһB

ϔұҵPҶӒFҷBҵҲҷҲKB A1 A2 ...An KF ҮҲҺFҴҼҺҲһB

ҬFҴPKFPҫҺBұҽKҽҹҺҲұҶBҼҲӁҷҽҹPҬҺӂһҽ ҭFҷFҺBҼҺҲһF ҲұҬPҮҷҲӀF ҼFҹPҬҺӂҲ

ϔұҬPҮҷҲӀFҴPKFҹҺPҵBұFҴҺPұҼүҶүҷBҬPҮҲӒFһҽ ҲҬҲӀF ҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲ B ӓFҷҮFPҲұҶFӋҽҮҬFһҽһFҮҷFҲҬҲӀFKF һҼҺBҷB ҲҵҲ ҹӒPһBҷ ҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲ

ҝҫҽҮҽӔFӔFҶPҹPҮҹҺҲұҶBҼҲӁҷPҶҹPҬҺӂҲҹPҮҺBұҽҶFҬBҼҲҲһҴӒҽӁҲҬPPҷF ӁҲKBKFҬPҮҲӒBұBҼҬPҺFҷBҹҺPһҼBҲұҵPҶӒFҷBҵҲҷҲKB ҼKҶҷPҭPҽҭBPҷBҵҲҷҲKB һҵ

ҬBҴBҺBҬBҷ β ҴPKBһFӁFKFҮҷҽҲұҬPҮҷҲӀҽ ҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲһFӁFһҬFӓFҷFҲұҬPҮҷҲ ӀF ҹBҲҲҬҲӀFϓBҼBҴҬҽҺBҬBҷҴBҰFҶPҮBKF

ҹҺFһFӁҷBҺBҬBҷ ҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲ ϛҺҲұҶBҼҲӁҷBҹPҬҺӂҺҪұҵҪҰүҷBҮҬF PҫҵBһҼҲһҴҽҹһҬҲҿҼBӁBҴBҹҺPһҼPҺBҴPKFKPKҷF ҹҺҲҹBҮBKҽϛҺҲҼPҶFһҽҽҲһҼPKPҫҵBһҼҲҮҬF ҼBӁҴFҴPKFҷFҹҺҲҹBҮBKҽҹPһҶBҼҺBҷPKҹҺҲұҶB ҼҲӁҷPKҹPҬҺӂҲ BҴPҲһBҶPBҴPһFҶPҭҽһҹPKҲ ҼҲҲұҵPҶӒFҷPҶҵҲҷҲKPҶҴPKBҷFҶBұBKFҮҷҲӁ ҴҲҿҼBӁBҴBһBҹҺҲұҶBҼҲӁҷPҶҹPҬҺӂҲ

ϛҺFһFҴҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲһBҷFҴPҶ ӓFҷPҶҹҺFһFӁҷPҶҺBҬҷҲKFҶҷPҭPҽҭBPҷҪҵҲ

ҷҲӑҪҴPKҪҲҶBPҷPҵҲҴPҼүҶүҷBҴPҵҲҴPҹҺҲұҶBҼҲӁҷBҹPҬҺӂҲҶBҲҬҲӀB0ҷB PҫҵBһҼ ҴPKBһBҮҺҰҲҽҷҽҼҺBӂӓPһҼҼҸҭҪҶҷPҭPҽҭҵBҷBұҲҬBһF ҽҷҽҼҺBӂӓBPҫҵBһҼ ҹҺҲұҶBҼҲӁҷFҹPҬҺӂҲ0ҷBҮҺҽҭBPҫҵBһҼKF һҹPӒBӂӓBPҫҵBһҼ ҹҺҲұҶBҼҲӁҷFҹP ҬҺӂҲ

ϛҺҲұҶBKFҭFPҶFҼҺҲKһҴPҼFҵPPҭҺBҷҲӁFҷPҹҺҲұҶBҼҲӁҷPҶҹPҬҺӂҲҲҮҬFҶB ҹBҺBҵFҵҷҲҶҹҺFһFӁҷҲҶҺBҬҷҲҶBҼFҹPҬҺӂҲ һҵ ҬBҴBPҮҮҬFҺBҬҷҲҲұҭPҺӓFҮFҾҲҷҲӀҲKFҲҶBһBҹҺҲұҶBҼҲӁҷPҶҹPҬҺӂҲ

ӑүҬҲҶҪҽҼҲҶҺҪҬҷҲҶҪ

BҴPҺBҬBҷ γ һBҮҺҰҲҮҬFҷFһҽһFҮҷFҫPӁҷFҲҬҲӀF

ҬFҼҺҲҬҺһҼFҹҺFһFҴBҮBҼFһҽҷBһҵҲӀҲҪ ҫ Ҭ

ҬBҷBPһҷPҬBҶBҮPҴBұҽKFһFҷBҲһҼҲҷBӁҲҷ

ύPӁҷFһҼҺBҷFҹҺBҬFҹҺҲұҶFһҽҹҺBҬPҽҭBPҷҲӀҲ ώҲһҲҷBҹҺBҬFҹҺҲұҶFKFҮҷBҴBKFҮҽҰҲҷҲҫPӁҷFҲҬҲӀF ϐүҾҲҷҲӀҲӑҪ

ҺҲұҶB ӁҲKBKFPһҷPҬBҹBҺBҵFҵPҭҺBҶ ҷBұҲҬBһFҹBҺBҵFҵPҹҲҹFҮ

ҬFӂҼPKFҺFӁFҷPұBҹҺҲұҶҽҽPҹӂҼF ҬBҰҲҲұBҹBҺBҵFҵPҹҲҹFҮ.FӋҽҼҲҶ ҹBҺBҵFҵPҹҲҹFҮҲҶBҲҷFҴBҹPһFҫҷBһҬPKһҼҬB

ҬBҴBһҼҺBҷBҹBҺBҵFҵPҹҲҹFҮBҶPҰFһFҽұFҼҲұBӓFҭPҬҽPһҷPҬҽ

ҺBҬҹBҺBҵFҵPҹҲҹFҮ ӁҲKBKFPһҷPҬBҹҺBҬPҽҭBPҷҲҴ ҷBұҲҬBһFҹҺBҬPҽҭҵҲ

ҝҭBPҲұҶFӋҽҫҲҵPҴPKFҮҬFһҽһFҮҷFҲҬҲӀFҴҬBҮҺBKFҹҺBҬ ϐҽҰҲҷFҼҺҲҽұBKBҶҷPҷPҺҶBҵҷFҲҬҲӀFҴҬBҮҺBһBұBKFҮҷҲӁҴҲҶҼүҶүҷPҶҷB ұҲҬBKҽһFҮҲҶFҷұҲKFҴҬBҮҺBo ҮҽҰҲҷB ӂҲҺҲҷBҲҬҲһҲҷB

ϖPӀҴBҲҶBһҬBһҬPKһҼҬBҴҬBҮҺB BҵҲҲҷFҴBҹPһFҫҷB ҬFҲҬҲӀFҴPӀҴFһҽҶFӋҽһPҫҷPҹPҮҽҮBҺҷF ϔһҹҲҼBKҶPKPӂҷFҴBһҬPKһҼҬBҴҬBҮҺBҲҴPӀҴF ϞүҸҺүҶҪϖҬBҮҺBҼҷBҮҮҲKBҭPҷBҵPҶ d ҴҬBҮҺBKFҮҷBҴKFұڈҲҺҽҴҬBҮҺBҼBҷBҮ

ӓFҭPҬҲҶҲҬҲӀBҶBҲұKFҮҷPҭҼүҶүҷB ҼK d2 = a 2 + b2 + c 2 ϐPҴBұ 5ҺPҽҭBP BDD1 KFҹҺBҬPҽҭҵҲ һҵ ҵ

BҲҶF ҹҺBҬB DD1 ҷPҺҶBҵҷBKFҷBҹҺBҬF DC Ҳ DA ҺBҬҷҲ

d2 = |BD1 |2

ϐBҫҲһFҷBҹҺBҬҲPҶPҮFҵҹҺҲұҶFPҮҴBҺҼPҷB ҲҵҲҹPһҽҮBҽPҫҵҲҴҽҹҺҲ ұҶFPҮҵҲҶB ҴPҺҲһҼҲһFҶҺFҰBҹҺҲұҶF һҵ 0ҮPҮҭPҬBҺBKҽӔFҭҶBҼFҺҲKBҵB ҲһFӁFһFҶҺFҰBҹҺҲұҶFҲҹҺFһBҬҲKBӓFҶҹPҲһҹҺFҴҲҮBҷҲҶҵҲҷҲKBҶB һҵ Ҳ ӒFҹӒFӓFҶ һҹBKBӓFҶ ҮPҫҲKBһFҶPҮFҵҹҺҲұҶFҲҵҲҹPһҽҮB

ϓ"ϐ"Ҡϔ

0ҮҺFҮҲҫҺPKҼүҶүҷB ҲҬҲӀBҲһҼҺBҷBӁFҼҬPҺPһҼҺBҷFҹҺҲұҶF ϖPKBҭFPҶFҼҺҲKһҴBҾҲҭҽҺBKFһҼҺBҷBҴPӀҴF

FҴBһҽ d1 ,d2 ,d3 ҮҽҰҲҷFҮҲKBҭPҷBҵBҼҺҲһҼҺBҷFҴҬBҮҺBһBұBKFҮҷҲӁҴҲҶҼүҶүҷPҶ 0ҮҺFҮҲҮҽҰҲҷҽҮҲKBҭPҷBҵFҴҬBҮҺBҽұBҬҲһҷPһҼҲPҮ d1 ,d2 Ҳ d3

ϐPҴBҰҲҮBһҽһҬFҮҲKBҭPҷBҵFҴҬBҮҺBҶFӋҽһPҫҷPҹPҮҽҮBҺҷF ϐBҼBKFҮҽҰҹPҮҽҮBҺҷBҮҲKBҭPҷBҵҲҴPӀҴFϖPҷһҼҺҽҲӂҲҮҽҰҹPҮҽҮBҺҷҽҲҬҲӀҲҼF ҴPӀҴF

ϐBҵҲҹҺFһFҴҹҺҲұҶFһBҷFҴPҶҺBҬҷҲҶPҰFҫҲҼҲҲһҼPҬҺFҶFҷPҷҸҺҶBҵBҷҲҹBҺBҵF ҵBҷ

ҺPҲұҬPӒҷBҺBҬBҷ α һFӁFӁFҼҲҺҲҲҬҲӀFҹBҺBҵFҵPҹҲҹFҮBҹPҽҷҽҼҺBӂӓҲҶҼBӁҴB ҶBϐPҴBҰҲҮBKFҹҺFһFҴҹBҺBҵFҵPҭҺBҶ

ϐҽҰҲҷFPһҷPҬҷҲҿҲҬҲӀBҹҺBҬPҭҹBҺBҵFҵPҹҲҹFҮBһҽDNҲDN5FҲҬҲӀFұBҴҵB ҹBKҽҽҭBPPҮ 60◦ BҫPӁҷBҲҬҲӀBKFӓҲҿPҬBһҺFҮӓBҹҺPҹPҺӀҲPҷBҵB0ҮҺFҮҲҮҽҰҲҷF ҮҲKBҭPҷBҵBҼPҭҹBҺBҵFҵPҹҲҹFҮB

0ҮҺFҮҲҹPҬҺӂҲҷFҮҲKBҭPҷBҵҷҲҿҹҺFһFҴBҴҬBҮҺBһBҲҬҲӀBҶBҮҽҰҲҷFDN DNҲ DN

ϖPӀҴBҲҬҲӀFNҹҺFһFӁFҷBKFһBҺBҬҷҲҴPKBһBҮҺҰҲҮҲKBҭPҷBҵҽҮPӓFPһҷPҬFҲҹҺP ҵBұҲҴҺPұһҺFҮҲӂҼFKFҮҷFPҮҭPҺӓҲҿPһҷPҬҷҲҿҲҬҲӀB һҵ 0ҮҺFҮҲҹPҬҺӂҲҷҽ ҹҺFһFҴB

0һҷPҬBҹҺBҬFҹҺҲұҶFһBҬҲһҲҷPҶ H KFKFҮҷBҴPҴҺBҴҼҺPҽҭBPһBPһҷPҬҲӀPҶ a ҲҴҺB ҴPҶ ҫPӁҷPҶһҼҺBҷҲӀPҶ bϖҺPұPһҷPҬҲӀҽҼPҭҼҺPҽҭҵBҲҬҺҿҮҺҽҭF ҭPҺӓF PһҷPҬF ҹҺҲұҶFҹPһҼBҬӒFҷBKFҺBҬBҷ B 0ҮҺFҮҲPҫҵҲҴҮPҫҲKFҷPҭҹҺFһFҴB ҫ 0ҮҺFҮҲҹPҬҺӂҲҷҽҹҺFһFҴBҽұҪҬҲһҷҸһҼҲҸҮ a b Ҳ H 0һҷPҬBҹҺBҬFҹҺҲұҶFKFҼҺPҽҭBPһBһҼҺBҷҲӀBҶBҮҽҰҲҷFDN DNҲDNϖҺPұ ҷBKҬFӔҽһҼҺBҷҲӀҽPһҷPҬFҲһҺFҮҲӂҼFҷBһҹҺBҶҷFҫPӁҷFҲҬҲӀFҹҺҲұҶFҹPһҼBҬӒFҷB KFҺBҬBҷϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҮPҫҲKFҷPҭҹҺFһFҴBBҴPKFҬҲһҲҷBҹҺҲұҶF H =30 DN

ϚһҷҸҬҪҹҺҪҬүҹҺҲұҶүӑүҺҸҶҫϐҽҰҲҷүҮҷӑҪҭҸҷҪҵҪҹҺҲұҶүһҽDNҲDNҪҬҲһҲҷҪ ӑүDNϚҮҺүҮҲҮҽҰҲҷҽҸһҷҸҬҷүҲҬҲӀү

Ϝ0 φ" Ϟ" 0ώϜҢ ϔϜ" ϔϐ"

PһҶBҼҺBKҶPҶҷPҭPҽҭBP A1 A2 ...An ҲҼBӁҴҽ V ҴPKBҷFҹҺҲҹBҮBҺBҬҷҲҼPҭ ҶҷPҭPҽҭҵBϖҺPұһҬBҴҽҼBӁҴҽұBҼҬPҺFҷFҲұҵPҶӒFҷFҵҲҷҲKF A1 A2 ...An ҹҺPҵBұҲ ҼBӁҷPKFҮҷBҹPҵҽҹҺBҬBһBҹPӁFҼҴPҶҽҼBӁҴҲ V һҵ

ϐүҾҲҷҲӀҲӑҪ

FҴBKF V ҼBӁҴBҴPKBҷFҹҺҲҹBҮBҺBҬҷҲҶҷPҭPҽҭҵB A1 A2 ...AnҝҷҲKBһҬҲҿ ҹPҵҽҹҺBҬBһBұBKFҮҷҲӁҴҲҶҹPӁFҼҴPҶ V ҴPKFһFҴҽұBҼҬPҺFҷҽҲұҵPҶӒFҷҽҵҲ

A1 A2 ...An ҷBұҲҬBһF n

ӀF BһҼҺBҷҲӀFPһҷPҬFһҽ PһҷPҬҷFҲҬҲӀF

PһҷPҬFҲҬҺҿBKF PҶPҼBӁҹҲҺBҶҲҮF

ҲҬҲӀFKF ҫPӁҷBһҼҺBҷB ҹӒPһBҷ ҹҲҺBҶҲҮF

ϜBһҼPKBӓFPҮҬҺҿBҹҲҺBҶҲҮFҮPҺBҬҷҲPһҷPҬFKF ҬҲһҲҷBҹҲҺBҶҲҮF һҵ ώҲһҲҷҽҹҲҺBҶҲҮFPҫҲӁҷPPҫүҵFҰBҬBҶPһB HώҲһҲҷPҶҹҲҺBҶҲҮFҷBұҲҬBҶPҲ ҮҽҰPҮҺFӋFҷҽҬҺҿPҶҹҲҺBҶҲҮFҲӓFҭPҬPҶPҺҼPҭPҷBҵҷPҶҹҺPKFҴӀҲKPҶҷBҺBҬBҷ

ҹҲҺBҶҲҮBҷBұҲҬBһFKPӂҲ

ύPӁҷFһҼҺBҷFҹҲҺBҶҲҮFһҽҼҺPҽҭҵPҬҲ BҴPҮҹҺBҬҲҵҷFҹҲҺBҶҲҮFҼҲҼҺPҽ

+FҮҷBҴPҲҬҲӁҷBҹҲҺBҶҲҮBKFҹҲҺBҶҲҮBӁҲKFһҽһҬFҲҬҲӀFҲһҼFҮҽҰҲҷF

ϛҺBҬBҴPKBҹҺPҵBұҲҴҺPұҬҺҿҹҺBҬҲҵҷFҹҲҺBҶҲҮFҲӀFҷҼBҺ ӀFҷҼBҺPҹҲ һBҷFҴҺҽҰҷҲӀF PһҷPҬFҷBұҲҬBһF PһB ҼFҹҲҺBҶҲҮF0һBҹҺBҬҲҵҷFұBҺҽҫӒFҷF ҹҲҺBҶҲҮFKFҹҺBҬBҴPKBҹҺPҵBұҲҴҺPұӀFҷҼҺFӓFҷҲҿPһҷPҬB ϓBҲұҺBҮҽҶPҮFҵBҲҵҲһҽҮBҽPҫҵҲҴҽҹҲҺBҶҲҮFҲҵҲұBҺҽҫӒFҷFҹҲҺBҶҲҮF ҴPҺҲһҼFһFPҮҭPҬBҺBKҽӔFҶҺFҰF BһҵҲӀҲҹҺFҮһҼBҬӒFҷBKFҶҺFҰBKFҮҷFҹҺB ҬҲҵҷFӁFҼҬPҺPһҼҺBҷFҹҲҺBҶҲҮF

BӀҺҼBKҼFҷBҹBҹҲҺҽҶҺFҰҽҹҺBҬҲҵҷFӁFҼҬPҺPһҼҺBҷFұBҺҽҫӒFҷFҹҲҺBҶҲ ҮF ҲұҺFҰҲҼFKFҲҷBҹҺBҬҲҼFPҮҭPҬBҺBKҽӔҲҶPҮFҵ

ҺҲҶүҺ "ҴPKFҹҲҺBҶҲҮBҹҺFһFӁFҷBһBҺBҬҷҲҴPKBKFҹBҺBҵFҵҷBPһҷPҬҲ ҮPҴB ҰҲҶPҮBҬBҰҲ

B ҼBҺBҬBҷҮFҵҲڈPӁҷFҲҬҲӀFҲҬҲһҲҷҽ ҷBҹҺPҹPҺӀҲPҷBҵҷFҮҽҰҲ

ڈ ҹҺFһFҴKFҶҷPҭPҽҭBPһҵҲӁBҷPһҷPҬҲ

Ҭ ҹPҬҺӂҲҷBҹҺFһFҴBҲPһҷPҬFPҮҷPһFһFҴBPҴҬBҮҺBҼҲӓҲҿPҬҲҿҺBһҼPKBӓBPҮ ҬҺҿBҹҲҺBҶҲҮF

ϜFӂFӓF ϐBKFҶPһBҶPһҴҲӀҽҮPҴBұBϐFҼBӒҲҮPҴBұBPһҼBҬӒBKҽһFұBҬFҰҫҽ B ϐPҬPӒҷPKFҹPһҶBҼҺBҼҲPҮҭPҬBҺBKҽӔFҽҭҵPҬFӁҲKҲһҽҴҺBӀҲҹҺFһFӁFҷҲҹBҺBҵFҵ ҷҲҶҹҺBҬBҶB һҵ ҲҹҺҲҶFҷҲҼҲ5BҵFһPҬҽҼүҸҺүҶҽ BҹҺҲҶFҺ ҴҺBӀҲҽҭҵB BVC ҹҺFһFӁFҷҲһҽҹҺBҬBҶB

ϓ"ϐ"Ҡϔ

FҴBKF a ҮҽҰҲҷBҲҬҲӀF H ҬҲһҲҷBҲ h BҹPҼFҶBKFҮҷBҴPҲҬҲӁҷFӁFҼҬPҺPһҼҺBҷF

ҹҲҺBҶҲҮF

B ҮBҼPKF a ҲұҺBӁҽҷBK H Ҳ h

ҫ ҮBҼPKF H ҲұҺBӁҽҷBK a Ҳ h

Ҭ ҮBҼPKF h ҲұҺBӁҽҷBK a Ҳ H

FҴBKF a ҮҽҰҲҷBPһҷPҬҷF b ҮҽҰҲҷBҫPӁҷFҲҬҲӀF h BҹPҼFҶBҲ H ҬҲһҲҷBҹҺBҬҲҵҷF

ӁFҼҬPҺPһҼҺBҷFҹҲҺBҶҲҮF0ҮҺFҮҲҷFҹPұҷBҼFPҮҼFӁFҼҲҺҲҬFҵҲӁҲҷFBҴPһҽҮBҼF ҬFҵҲӁҲҷF

B a,bҫ a,hҬ a,H

ҭ b,hҮ b,HӋ h,H

"ҴPKFҹҲҺBҶҲҮBҹҺFһFӁFҷBKFҮҷPҶҺBҬҷҲҹBҺBҵFҵҷPPһҷPҬҲҼBҴPҮBKFҬҲһҲҷBҹҺF ҹPҵPҬӒFҷB PҮҺFҮҲPҮҷPһҹPҬҺӂҲҷBҹҺFһFҴBҲPһҷPҬF

BҴPҶҺBһҼPKBӓҽPҮҬҺҿBҹҲҺBҶҲҮFҬҲһҲҷF H ҼҺFҫBҹҺFһFӔҲҼҽҹҲҺBҶҲҮҽKFҮҷPҶ ҺBҬҷҲҹBҺBҵFҵҷPPһҷPҬҲ ҹBҮBҹPҬҺӂҲҷBҹҺFһFҴBҫҽҮFKFҮҷBҴBҹPҵPҬҲҷҲҹPҬҺ ӂҲҷFPһҷPҬF

ҺBҬҲҵҷBӂFһҼPһҼҺBҷBҹҲҺBҶҲҮBҬҲһҲҷFDNҲҮҽҰҲҷFPһҷPҬҷFҲҬҲӀFDN ҹҺFһFӁFҷBKFKFҮҷPҶҺBҬҷҲҹBҺBҵFҵҷPPһҷPҬҲϔұҺBӁҽҷBKҺBһҼPKBӓFҹҺFһFӁҷFҺBҬ

ҷҲPҮҬҺҿBҹҲҺBҶҲҮFBҴPKFҹPҬҺӂҲҷBҹҺFһFҴB

0һҷPҬFҮҬFҹҲҺBҶҲҮFҲҶBKҽKFҮҷBҴFҹPҬҺӂҲҷFҲҵFҰFҽҲһҼPKҺBҬҷҲҮPҴKFҬҲһҲҷB KFҮҷFҼҺҲҹҽҼBҬFӔBPҮҬҲһҲҷFҮҺҽҭF0ҮҺFҮҲPҮҷPһҹPҬҺӂҲҷBҹҺFһFҴBҼFҮҬFҹҲ ҺBҶҲҮFһBKFҮҷPҶҺBҬҷҲҴPKBKFҹBҺBҵFҵҷBPһҷPҬBҶBҷBҺBһҼPKBӓҽ 2 3

ҶBӓFҬҲһҲҷF

0һҷPҬBҹҲҺBҶҲҮFKFҹҺBҬPҽҭBPҷҲҴһBһҼҺBҷҲӀBҶBҮҽҰҲҷFDNҲ DNώҲһҲ ҷBKFҹPҮFӒFҷBҷBҼҺҲҹPҮҽҮBҺҷBҮFҵBҲҴҺPұҼBӁҴFҹPҮFҵFһҽҹPһҼBҬӒFҷFҺBҬҷҲ ҹBҺBҵFҵҷFPһҷPҬҲ0ҮҺFҮҲҹPҬҺӂҲҷFҹҺFһFҴB

ҪҴҸҶҺҪһҼҸӑҪӓҽҸҮҬҺҿҪҹҲҺҪҶҲҮүһҪҬҲһҲҷҸҶ

ҺҲұҶBҲҹҲҺBҶҲҮBһҽҹҺҲҶFҺҲҹPҵҲFҮBҺBϜBұҶPҼҺҲӔFҶPҲҷFҴFҮҺҽҭF ҹPҵҲFҮҺF

ϓBҮҬBҶҷPҭPҽҭҵBҴBҰFҶPҮBһҽ һҽһFҮҷҲ BҴPKFӓҲҿPҬҹҺFһFҴKFҮҷBұBKFҮ ҷҲӁҴBһҼҺBҷҲӀBҼBҮҬBҶҷPҭPҽҭҵB ϐүҾҲҷҲӀҲӑҪ

PҵҲFҮBҺһҴBҹPҬҺӂKFҽҷҲKBҴPҷBӁҷPҭһҴҽҹBҶҷPҭPҽҭҵPҬBҴPKҲұBҮPҬPӒB ҬBKҽһҵFҮFӔFҽһҵPҬF

ϓBһҬBҴBҮҬBҷFһҽһFҮҷBҶҷPҭPҽҭҵB M Ҳ M ҲұҼPҭһҴҽҹBҹPһҼPKҲҷҲұ

ҶҷPҭPҽҭҵPҬB M,M1 ,...,Mn,M PҫҺBұPҬBҷPҮҶҷPҭPҽҭҵPҬBҼPҭBһҴҽҹB ҼBҴPҮBһҽһҬBҴBҮҬBҽұBһҼPҹҷBҶҷPҭPҽҭҵBҽҷҲұҽһҽһFҮҷҲ ϓBKFҮҷҲӁҴBҲҬҲӀBһҬBҴBҮҬBһҽһFҮҷBҶҷPҭPҽҭҵBҷҲKFҲҬҲӀBҷҲKFҮ ҷPҭҮҺҽҭPҭҶҷPҭPҽҭҵB

ϐҬBҷFһҽһFҮҷBҶҷPҭPҽҭҵBһҽҲҵҲҮҲһKҽҷҴҼҷҲҲҵҲҲҶKFҹҺFһFҴӑүҮҷҸ ұҪӑүҮҷҲӁҴҸҼүҶүҼҪҮҬҪҶҷPҭPҽҭҵB

ҹPҬҺӂҲ

PҵҲFҮBҺһҴBҹPҬҺӂKF ұBҼҬPҺFҷB BҴPKFһҬBҴB

ӓFҷBҲҬҲӀBұBKFҮҷҲӁҴBһҼҺBҷҲӀBҮҬFһҽһFҮҷFһҼҺB ҷF BҽҹҺPҼҲҬҷPҶKF PҼҬPҺFҷB

PҬҺӂҲҹҺҲұҶҲҲҹҲҺBҶҲҮBһҽҹҺҲҶFҺҲұB ҼҬPҺFҷҲҿҹPҵҲFҮBҺһҴҲҿҹPҬҺӂҲ

BһҵҲӀҲҹҺҲҴBұBҷKFҹҺҲҶFҺKFҮҷFPҼҬPҺF ҷFҹPҵҲFҮFҺһҴFҹPҬҺӂҲ

ϓBҼҬPҺFҷBҹPҵҲFҮBҺһҴBҹPҬҺӂҺBұҵBҰFҷBҮҬF PҫҵBһҼҲһҴҽҹһҬҲҿҼBӁBҴBҹҺPһҼPҺBҴPKFҷFҹҺҲҹB ҮBKҽҼPKҹPҬҺӂҲ+FҮҷBPҮӓҲҿKFPҭҺBҷҲӁFҷBҲҷBұҲҬBҶPKF

KFҷFPҭҺBҷҲӁFҷBҲҷBұҲҬBҶPKF

ҺFҷFҭPӂҼPһҶPҮFҾҲҷҲһBҵҲҹPҵҲFҮBҺҽұFҵҲһҶPҹPKҶPҬFҹPҵҲFҮBҺһҴFҹPҬҺӂҲ

ҭҵFҮBҮPһҼBһҵPҰFҷB BҼBKҷBӁҲҷ ҶFӋҽҼҲҶ ҹPһҼҲҰFҶPҮBҮFҾҲҷҲӀҲKPҶҹPҵҲFҮҺB ҴBPҮFҵBҹҺPһҼPҺBPҭҺBҷҲӁFҷPҭұBҼҬPҺFҷPҶҹPҵҲFҮBҺһҴPҶҹPҬҺӂҲPҫҽҿҬBҼҲҶPҼBӁҷP PҷPӂҼPKFҽһҴҵBҮҽһBҷBӂPҶҲҷҼҽҲҼҲҬҷPҶҹҺFҮһҼBҬPҶPҹPҵҲFҮҺҽ ύҲҼҷBPһPҫҲҷBҹPҵҲFҮҺBKFҮBKFҼPҼFҵPӁҲKBKFҹPҬҺӂһBһҼBҬӒFҷBҲұҶҷPҭPҽҭҵPҬB "ҹһҼҺBҿҽKҽӔҲӓFҭPҬҽҶBҼFҺҲKBҵҷPһҼ ҼPKFҮFPҹҺPһҼPҺBPҭҺBҷҲӁFҷҴPҷBӁҷҲҶҫҺPKFҶ ҶҷPҭPҽҭҵPҬB ҺFҶBҼPҶF ҼPKFPҭҺBҷҲӁFҷҮFPҹҺPһҼPҺBϐҺҽҭPҫҲҼҷPһҬPKһҼҬPKFҮBKF ҼBKҮFPҹҺPһҼPҺBҹPҬFұBҷ ӁBҴҲBҴPһFPҭҺBҷҲӁҲҶPҷBӓFҭPҬҽҽҷҽҼҺBӂӓPһҼ ҼKһҬB ҴFҮҬFӓFҭPҬFҼBӁҴFҶPҭҽһFһҹPKҲҼҲҲұҵPҶӒFҷPҶҵҲҷҲKPҶҴPKBӀFҵBҵFҰҲҽҼPҶҮFҵҽ ҹҺPһҼPҺB

BҹҺҲҶFҺҲҶBҹҺҲұҶҲҲҹҲҺBҶҲҮBҵBҴPҹҺPҬFҺBҬBҶPҮBӓҲҿPҬFҹPҬҺӂҲұBҮPҬP

ҴPKFҲҷҼҽҲҼҲҬҷPҷFҹҺҲҿҬBҼBҶPҴBPҹPҵҲFҮҺF ҷBҹҺҲҶFҺ

oҮҬFҹPҮҽҮBҺҷFҴPӀҴFӁҲKҲKFҹҺFһFҴһBҶPKFҮҷBӓҲҿPҬBұBKFҮҷҲӁҴBҲҬҲӀB ҷҲKF ұBҮPҬPӒFҷҽһҵPҬ

oҮҬFҹҲҺBҶҲҮFӁҲKBKFKFҮҲҷBұBKFҮҷҲӁҴBҼBӁҴBӓҲҿPҬұBKFҮҷҲӁҴҲҬҺҿ ҷҲKFұB ҮPҬPӒFҷҽһҵPҬ

oҼFҵPҴPKFһFҮPҫҲKFҴBҮһFҲұҴPӀҴFҲһFӁFҹҲҺBҶҲҮBӁҲKBһFPһҷPҬBһBKFҮҷPҶ һҼҺBҷPҶҴPӀҴF BҬҺҿKPKKFҽӀFҷҼҺҽҷBһҹҺBҶҷFһҼҺBҷFҴPӀҴF ҷҲKFұBҮPҬPӒFҷ ҽһҵPҬ

ϞүҸҺүҶҪ

ҴPҶڈҺPKҽ ӓFҭPҬҲҿҲҬҲӀB

ϐҸҴҪұ ҬBҴBһҼҺBҷB ҹӒPһBҷ ҹPҵҲFҮҺBҲҶBҲһҼPҼPҵҲҴP ҽҭҵPҬBҴPҵҲҴPҲһҼҺB ҷҲӀBϓBҼPKFڈҺPK ҽҭҵPҬBһҬҲҿҹӒPһҷҲҹPҵҲFҮҺBKFҮҷBҴұڈҲҺҽ ڈҺPKFҬBһҼҺBҷҲӀBһҬҲҿ ҹӒPһҷҲϖBҴPKFһҬBҴBҲҬҲӀBҹPҵҲFҮҺBһҼҺBҷҲӀBҮҬFһҽһFҮҷFҹӒPһҷҲ ҼPKF ҽ ҹPҶFҷҽ ҼPҶұڈҲҺҽ һҬBҴBһҼҺBҷҲӀBҺBӁ

0ҮҺFҮҲһFҹҺҬPұҫҲҺҽҷҽҼҺBӂӓҲҿҽҭҵPҬBҶҷP

ҭPҽҭҵB AB...F ҴPKҲPҫҺBұҽKFҴPҷҼҽҺҽҹҺPKFҴӀҲKF

ҹPҵҲFҮҺB"ҴPKF k ҫҺPKһҼҺBҷҲӀBҼPҭҶҷPҭPҽҭҵB ұҫҲҺӓFҭPҬҲҿҽҷҽҼҺBӂӓҲҿҽҭҵPҬBKF (k 2) 180◦ 5P KFұҫҲҺҽҭҵPҬBһBҼүҶүҷҲҶBҷBҴPҷҼҽҺҲ ҴPKҲPҮҭP ҬBҺBKҽҲҬҲӁҷҲҶҽҭҵPҬҲҶBһBҭPҺӓF ҹҺFҮӓF һҼҺBҷF ҹPҵҲFҮҺBϔһҼPҼPҵҲҴPҲұҷPһҲҲұҫҲҺҽҭҵPҬBһBҼү ҶүҷҲҶBҷBҴPҷҼҽҺҲ ҴPKҲPҮҭPҬBҺBKҽҲҬҲӁҷҲҶҽҭҵP ҬҲҶBһBҮPӓFһҼҺBҷFҹPҵҲFҮҺBϐBҴҵF ҽҴҽҹBҷұҫҲҺ ҼBҴҬҲҿҽҭҵPҬBKF 2(k 2)·180◦ =(k 2)·360◦ 5PҶF

ұҫҲҺҽҼҺFҫBҮPҮBҼҲұҫҲҺҽҷҽҼҺBӂӓҲҿҽҭҵPҬBҶҷP ҭPҽҭҵPҬB5ҲҽҭҵPҬҲҲҶBKҽҼүҶүҷBҽҽҷҽҼҺBӂӓPһҼҲ ҴPҷҼҽҺF AB...F ϓҫҲҺһҬҲҿҽҭҵPҬBһBKFҮҷҲҶҼBҴҬҲҶұBKFҮҷҲӁҴҲҶҼүҶүҷPҶKF 360◦ ύҺPKҼBҴҬҲҿҼүҶүҷBKF n k ҹBKFҽҴҽҹBҷұҫҲҺҼBҴҬҲҿҽҭҵPҬB

KFҼҺBҰFҷҲҽҴҽҹBҷұҫҲҺ

KFҺKFڈҺPKҼүҶүҷB n +1

0ҬPһFҶPҰFҮPҫҲҼҲҲҷFҹPһҺFҮҷҲҶұBҴӒҽӁҲҬBӓFҶϙBҲҶF n һҼҺBҷBҹҲ

n · 180◦ 0һҷPҬBKF nҼPҽҭBP ҲұҫҲҺӓFҭPҬҲҿҽҭҵPҬBKF (n 2) · 180◦

ϐҸҴҪұ ϙFҴBKFڈҺPKҼүҶүҷBҹPҵҲFҮҺB

sϙFҴBҶҷP ҭPҽҭҵPҬҲҴPKҲPڈҺBұҽKҽ

ϖPҷҬFҴһBҷҹPҵҲFҮBҺKFҹҺBҬҲҵBҷBҴPһҽһҬFӓFҭPҬFһҼҺBҷҲӀFҹҺBҬҲҵҷҲ

ҬҺһҼBҹҺBҬҲҵҷҲҿҹPҵҲFҮBҺB

ϐҸҴҪұ ϖBPҹPһҵFҮҲӀBҼүҸҺүҶFһҵFҮҲҮBKFұڈҲҺҲҬҲӁҷҲҿ

ڈҲӔF 1 k 3 ҹBKFһPڈұҲҺPҶҷB 1 d > 1 2 1 3 = 1 6 ҼK d< 6ϔһҼPҼBҴPҮPڈҲKBҶP ұڈPҭ d 3 ҮBKF k< 6 ҺFҶBҼPҶF ҹPһҼPKҲһҵFҮFӔҲҿҹFҼҶPҭҽӔҷPһҼҲ

k =3 n =3

k =3 n =4

k =3 n =5

k =4 n =3

k =5 n =3

.PҰFһFҮPҴBұBҼҲFҾFҴҼҲҬҷPҶҴPҷһҼҺҽҴӀҲKPҶ ҮPҴBұҲұPһҼBҬӒBҶPұڈPҭҮҽҰҲҷF ҮBұBһҬBҴҲPҮҹFҼҷBҬFҮFҷҲҿһҵҽӁBKFҬBұBҲһҼBҲҹPһҼPKҲPҮҭPҬBҺBKҽӔҲҹҺBҬҲҵBҷҹP ҵҲFҮBҺ ҬҲһҽ PҷҲҴPҷҬFҴһҷFҾҲҭҽҺү ϐBڈҲһҶҸ ҽ һҬBҴPҶPҮPҬҲҿһҵҽӁBKFҬBPҮҺFҮҲҵҲڈҺҸӑҼүҶүҷB n ڈҺPKҹӒPһҷҲ s

ks =2m,dn =2m,n + s = m +2

BҼBKҷBӁҲҷҮPڈҲKBҶPһҬҲҿҹFҼҹҺBҬҲҵҷҲҿҹPҵҲFҮBҺBһBڈҺPKFҶҼүҶүҷB ҹӒPһҷҲҲҲҬҲ ӀB PҮBӀҲһҽ ҮBҼҲҹҺFҭҵFҮҷP ҽ һҵFҮFӔPKҼBڈFҵҲ BҹPҵҲFҮҺҲһҽ ҹҺFҮһҼBҬӒFҷҲҷBһҵҲ ӀҲ ✷

ϓ"ϐ"Ҡϔ

k d n s m BұҲҬҹҺBҬҲҵҷPҭҹPҵҲFҮҺB

ҼFҼҺBFҮBҺ

PҴҼBFҮBҺ

ҲҴPһBFҮBҺ

ҿFҴһBFҮBҺ ҴPӀҴB ҮPҮFҴBFҮBҺ

nҼPһҼҺBҷFҹҺҲұҶF

ϐҬBҹҺBҬҲҵҷBҼFҼҺBFҮҺBһҽһBһҼBҬӒFҷBҼBҴPҮBһFһҼҺBҷBKFҮҷPҭҹPҴҵBҹBһBһҼҺB ҷPҶҮҺҽҭPҭϐBҵҲKFҷBҼBKҷBӁҲҷҮPҫҲKFҷҲҹPҵҲFҮBҺҹҺBҬҲҵBҷ

ϐPҴBҰҲҮBһҽӀFҷҼҺҲһҼҺBҷBҴPӀҴFҼүҶүҷBKFҮҷPҭҹҺBҬҲҵҷPҭPҴҼBFҮҺB

ϐPҴBҰҲҮBһҽӀFҷҼҺҲ ҼFҰҲӂҼB һҼҺBҷBҹҺBҬҲҵҷPҭPҴҼBFҮҺBҼүҶүҷBKFҮҷFҴPӀҴF

ϐPҴBҰҲҮBһҽҼFҰҲӂҼBһҼҺBҷBҹҺBҬҲҵҷPҭҼFҼҺBFҮҺBҼүҶүҷBҮҺҽҭPҭҹҺBҬҲҵҷPҭҼF ҼҺBFҮҺB

ҬүӁүҼҲҺҲҬҲһҲҷүҹҺҪҬҲҵҷҸҭҼүҼҺҪүҮҺҪӑүҮҷҪҴүһҽҲһүҴҽһүҽҲһҼҸӑҼҪӁҴҲϞҪ ҼҪӁҴҪҮүҵҲһҬҪҴҽҬҲһҲҷҽҽҺҪұҶүҺҲ 3:1 ҺҪӁҽҷҪӑҽӔҲҸҮҼүҶүҷҪϐҸҴҪҰҲ

BҭҵBһҷPҮFҾҲҷҲӀҲKҲ һҬBҴҲҹPҵҲFҮBҺKFPҭҺBҷҲӁFҷҶҷPҭPҽҭҵPҬҲҶB

0ұҷBӁҲҶPһB P ҹPҬҺӂҲҷҽҹҺҲұҶF һB

P =2B + M.

ϞүҸҺүҶҪ PҬҺӂҲҷBPҶPҼBӁBڈҲҵPҴPKFҹҺҲұҶFKFҮҷBҴBKFҹҺPҲұҬPҮҽ PڈҲ ҶBҷPҺҶBҵҷPҭҹҺFһFҴBҲҮҽҰҲҷFӓFҷFڈPӁҷFҲҬҲӀF

ϐPҴBұ FҴBKF A1 A2 ...AnA 1 A 2 ...A n ҹҺҲұҶBҲ B1 B2 ...Bn ӓFҷҷPҺҶBҵҷҲҹҺFһFҴPҫҲҶB p BһҵҲ ӀҲҹҺFҮһҼBҬӒFҷKFһҵҽӁBK n =5 FҴBKF b ҮҽҰҲ ҷBҫPӁҷFҲҬҲӀFϐBҫҲһҶPҮPҫҲҵҲҹPҬҺӂҲҷҽPҶPҼB ӁB ҼҺFҫBҮBһBҫFҺFҶPҹPҬҺӂҲҷFҫPӁҷҲҿһҼҺBҷB ҬF ҫPӁҷFһҼҺBҷFһҽҹBҺBҵFҵPҭҺBҶҲ BһҼҺBҷҲӀFҷPҺҶBҵ ҷPҭҶBҵPҭҹҺFһFҴBһҽӓҲҿPҬFҬҲһҲҷF 0ҫҺBұҵPҰҲ ϓBҼPҲҶBҶP

M = s · b,

ҭҮFKF s PҫҲҶҷPҺҶBҵҷPҭҹҺFһFҴBҹҺҲұҶF ✷

"ҴPKFҹҺҲұҶBҹҺBҬB ҮҽҰҲҷBҫPӁҷFҲҬҲӀFKFҮҷBҴBKFҬҲһҲҷҲ BҷPҺҶBҵҷҲ

ҹҺFһFҴKFҶҷPҭPҽҭBPҹPҮҽҮBҺBҷPһҷPҬҲҹҺҲұҶF 0ҫҺBұҵPҰҲҼҲ ϐBҴҵF b = H Ҳ s = p ҭҮFKF p PҫҲҶPһҷPҬF ҹBKFҹPҬҺӂҲҷBPҶPҼBӁBҹҺBҬFҹҺҲұҶF M = p · H

ҲҺBҶҲҮBKFҹPҵҲFҮBҺϓBҼPKFӓFҷBҹP ҬҺӂҲҷBKFҮҷBҴBұҫҲҺҽҹPҬҺӂҲҷBҶҷPҭPҽҭҵPҬB

ҴPKҲKFPҭҺBҷҲӁBҬBKҽ

PҬҺӂҲҷBPҶPҼBӁBҹҲҺBҶҲҮFKFҮҷBҴBKF ұҫҲҺҽҹPҬҺӂҲҷBһҬҲҿӓFҭPҬҲҿҫPӁҷҲҿһҼҺBҷB ҼK

M = PBCV + PCDV + ··· + PABV һҵ

PҬҺӂҲҷBҹҲҺBҶҲҮF ҵ

ϞүҸҺүҶҪ PҬҺӂҲҷBPҶPҼBӁBҹҺBҬҲҵҷF

ҹҲҺBҶҲҮFKFҮҷBҴBKFҹPҵҽҹҺPҲұҬPҮҽ

PһҷPҬҲӀFҲҮҽҰҲҷFBҹPҼFҶF

ϐPҴBұ FҴBKF VBC...A ҹҺBҬҲҵҷB nҼPһҼҺBҷBҹҲҺBҶҲҮB һҵ FҴBKF |BC| = |CD| = ··· = |AB| = a, (VN )⊥(CD), |VN | = h oҮҽҰҲҷBBҹPҼFҶFύPӁҷFһҼҺBҷFҹҲҺBҶҲҮFһҽҹPҮҽҮBҺҷҲKFҮҷBҴPҴҺBҴҲ ҼҺPҽҭҵPҬҲ PҬҺӂҲҷBKFҮҷPҭҼҺPҽҭҵBKF

PҬҺӂҲҷFҹPҵҲFҮBҺB

ҷBҹҲҺBҶҲҮFҹҺPҵBұҲҴҺPұҹҺFһFӁҷ

ҹPҬҺӂҲҷҽ ҹҲҺBҶҲҮF

ϜFӂFӓF һҵ M = PΔBCV + PΔCDV + PΔDAV + PΔABV .

.FӋҽҼҲҶ ΔBCV ∼ = ΔCDV ∼ = ΔDAV ∼ = ΔABV,

ҹBKFҹҺFҶBҼPҶF M =4PΔBCV

ҭBP BOC KFҹҺBҬPҽҭҵҲһBҴBҼFҼBҶB |OB| =3 N

|OC| =4 N ҹBKF |BC| = 32 +42 =5 NϓBҼBK

ҹҺBҬPҽҭҵҲҼҺPҽҭBPҼBҴPӋFҬBҰҲ

BPһҷPҬҽPһPҫҲҷBҮҲKBҭPҷBҵBҺPҶҫB ҼҺPҽ ҵ

PҬҺӂҲҷBPҶPҼBӁBұBҺҽҫӒFҷFҹҲҺBҶҲҮFKFҮҷBҴBKFұҫҲҺҽҹPҬҺӂҲҷBһҬҲҿ ӓFҷҲҿҫPӁҷҲҿһҼҺBҷB

ϞүҸҺүҶҪ PҬҺӂҲҷBPҶPҼBӁBҹҺBҬҲҵҷFұBҺҽڈӒFҷFҹҲҺBҶҲҮFKFҮҷBҴBKF ҹҺPҲұҬPҮҽ ҹPҵҽұڈҲҺBPڈҲҶBҮҬFӓFҷFPһҷPҬFҲҮҽҰҲҷFBҹPҼFҶF

ϐPҴBұ FҴBKF BC...AB1 C1 ...A1 ҹҺBҬҲҵҷBұBҺҽҫӒFҷBҹҲҺBҶҲҮB һҵ FҴBKFҮBӒF |BC| = |CD| = = |AB| = a, |B1 C1 | = |C1 D1 | = ··· = |A1 B1 | = a1 ,

p = na oPҫҲҶҮPӓFPһҷPҬF

p1 = na1 oPҫҲҶҭPҺӓFPһҷPҬF

(NN1 )⊥(BC) |NN1 | = h oҮҽҰҲҷBBҹPҼFҶF

ύPӁҷFһҼҺBҷFһҽҹPҮҽҮBҺҷҲKFҮҷBҴP

ҴҺBҴҲҼҺBҹFұҲ ϓBӂҼP PҬҺӂҲҷBKFҮҷPҭ ҼҺBҹFұBKF PBCC1 B1 = a + a1 2 h.

PҬҺӂҲҷBPҶPҼBӁBKF n ҹҽҼBҬFӔB ҼK M = n a + a1 2 h = na + na1 2 h = p + p1 2 h; M = p + p1 2 h ✷

Ҭ M =150 EN2 h =15 EN p =12 EN p1 = 2M h p = 2 150 15 12=8 EN

PҬҺӂҲҷBұBҺҽڈӒFҷFҹҲҺBҶҲҮFҮPڈҲKBһFҴBҮһFҹPҬҺӂҲҷҲPҶPҼBӁBҮPҮBKҽ ҹPҬҺ

ӂҲҷFPڈFӓFҭPҬFPһҷPҬF ҼK

P = M + B + B1

ϓ"ϐ"Ҡϔ

0ҮҺFҮҲҹPҬҺӂҲҷҽҹҺBҬҲҵҷF

B ҼҺPһҼҺBҷF

ҫ ӁFҼҬPҺPһҼҺBҷF

Ҭ ӂFһҼPһҼҺBҷF

ҹҺҲұҶFһBҲҬҲӀPҶҮҽҰҲҷFDNҲҬҲһҲҷPҶDN

0ҮҺFҮҲҹPҬҺӂҲҷҽҹҺBҬҲҵҷF

B ҼҺPһҼҺBҷF

ҫ ӁFҼҬPҺPһҼҺBҷF

Ҭ ӂFһҼPһҼҺBҷF

ҹҲҺBҶҲҮFһBPһҷPҬҷPҶҲҬҲӀPҶҮҽҰҲҷFDNҲҬҲһҲҷPҶDN

ϖPҵҲҴPҹҽҼBKFҬFӔBҹPҬҺӂҲҷBKFҮҷFҴPӀҴFPҮҹPҬҺӂҲҷFҮҺҽҭFҴPӀҴFBҴPKFҲҬҲӀB ҹҺҬFҮҬBҹҽҼBҮҽҰBPҮҲҬҲӀFҮҺҽҭF

ϐҽҰҲҷBPһҷPҬҷҲҿҲҬҲӀBҹҺBҬPҭҹBҺBҵFҵPҹҲҹFҮBһҽDNҲDNҝҭBPҲұҶFӋҽҼҲҿ ҲҬҲӀBKF 30◦ BҮҽҰҲҷBҫPӁҷFҲҬҲӀFKFDNϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҹBҺBҵFҵPҹҲҹFҮB

0ҮҺFҮҲҹPҬҺӂҲҷҽҹҺBҬҲҵҷFӁFҼҬPҺPһҼҺBҷFҹҺҲұҶFBҴPKFҮҽҰҲҷBӓFҷFҮҲKBҭP ҷBҵF d =14 DN BҮҽҰҲҷBҮҲKBҭPҷBҵFӓFҷFҫPӁҷFһҼҺBҷF d1 =10 DN

0һҷPҬBҹҺBҬFҹҺҲұҶFKFҺPҶҫһBҮҲKBҭPҷBҵBҶBҮҽҰҲҷFDNҲDN ϔұҺBӁҽҷBK ҹPҬҺӂҲҷҽҹҺҲұҶFBҴPҮҲKBҭPҷBҵBӓFҷFҫPӁҷFһҼҺBҷFҲҶBҮҽҰҲҷҽDN

0һҷPҬҷFҲҬҲӀFҹҺBҬPҭҹBҺBҵFҵPҹҲҹFҮBҲҶBKҽҮҽҰҲҷFDNҲDN KFҮҷBҮҲ KBҭPҷBҵBPһҷPҬFҲҶBҮҽҰҲҷҽDNBҬFӔBҮҲKBҭPҷBҵBҹBҺBҵFҵPҹҲҹFҮBKFDN ϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҹBҺBҵFҵPҹҲҹFҮB

0һҷPҬBҹҲҺBҶҲҮFKFҴҬBҮҺBҼһBһҼҺBҷҲӀPҶҮҽҰҲҷFEN+FҮҷBҫPӁҷBҲҬҲӀBҷPҺ ҶBҵҷBKFҷBҺBҬBҷPһҷPҬFBҬҲһҲҷBҹҲҺBҶҲҮFKFEN0ҮҺFҮҲҹPҬҺӂҲҷҽPҶPҼBӁB ҹҲҺBҶҲҮF

0һҷPҬҷBҲҬҲӀBҹҺBҬҲҵҷFӂFһҼPһҼҺBҷFҹҲҺBҶҲҮFҲҶBҮҽҰҲҷҽ a PҬҺӂҲҷBKFҮ ҷFҫPӁҷFһҼҺBҷFKFҮҷBҴBKFҹPҬҺӂҲҷҲҷBKҬFӔFҭҮҲKBҭPҷBҵҷPҭҹҺFһFҴBҹҲҺBҶҲҮF

ϚһҷҸҬҷҪҲҬҲӀҪҹҺҪҬҲҵҷүӁүҼҬҸҺҸһҼҺҪҷүҹҲҺҪҶҲҮүҲҶҪҮҽҰҲҷҽ a ҸҬҺӂҲҷҪҮҲ ӑҪҭҸҷҪҵҷҸҭҹҺүһүҴҪҼүҹҲҺҪҶҲҮүӑүҮҷҪҴҪӑүҹҸҬҺӂҲҷҲҸһҷҸҬүϚҮҺүҮҲҹҸҬҺӂҲҷҽ ҸҶҸҼҪӁҪҹҲҺҪҶҲҮү

ϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҹҺBҬҲҵҷFӂFһҼPһҼҺBҷFҹҲҺBҶҲҮFBҴPKFBҹPҼFҶBҹҲҺBҶҲҮF KFҮҷBҴB h BBҹPҼFҶBPһҷPҬF t

ϔұҺBӁҽҷBKҹPҬҺӂҲҷҽPҶPҼBӁBҹҺBҬҲҵҷFҮFһFҼPһҼҺBҷFҹҲҺBҶҲҮFBҴPKFҹPҵҽҹҺFӁ ҷҲҴҴҺҽҰҷҲӀFPҴPPһҷPҬF r BҬҲһҲҷBҹҲҺBҶҲҮFҮҽҰBPҮҼPҭҹPҵҽҹҺFӁҷҲҴBұB ҹPҵPҬҲҷҽPһҷPҬҷFҲҬҲӀF

0ҮҺFҮҲҹPҬҺӂҲҷҽҹҺBҬҲҵҷPҭ B ҼFҼҺBFҮҺB ҫ PҴҼBFҮҺB Ҭ ҿFҴһBFҮҺB

һBҲҬҲӀPҶҮҽҰҲҷF a

ҺBҬҲҵҷBӁFҼҬPҺPһҼҺBҷBұBҺҽҫӒFҷBҹҲҺBҶҲҮBҲҶBҬҲһҲҷҽ H ҲBҹPҼFҶҽ hϐҽҰҲ ҷFPһҷPҬҷҲҿҲҬҲӀBһҽ a Ҳ c BҫPӁҷF bϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҹҲҺBҶҲҮFBҴPKF B a =8 c =2 b =5 ҫ a =9 c =3 h =5 Ҭ a =13

ϔұҽӁBҬBKҽӔҲҶBҼFҶBҼҲҴҽӁFһҼPһҶPҶFҺҲҵҲҲҲұҺBӁҽҷBҬBҵҲҮҽҰҲҷFҲҹP ҬҺӂҲҷF BҮBӔFҶPҺFӂBҬBҼҲҹҺPҫҵFҶPҮҺFӋҲҬBӓBұBҹҺFҶҲҷFҷFҴҲҿҭFPҶFҼҺҲ

KһҴҲҿҼFҵB ҽҹҺҬPҶҺFҮҽҹPҵҲFҮBҺB

ҺҲPҮҺFӋҲҬBӓҽҮҽҰҲҷF ҺBһҼPKBӓB ҴPҺҲһҼҲҵҲһҶPҷFҴҽKFҮҲҷҲӀҽұBҶF ҺFӓFҮҽҰҲҷFo e ҶFҼBҺoN ӀFҷҼҺҲҶFҼBҺoDN ҶҲҵҲҶFҼBҺoNNҲҼҮ ϖBP

KFҮҲҷҲӀҽұBҶFҺFӓFҹPҬҺӂҲҷFҽұҲҶBҵҲһҶPҹPҬҺӂҲҷҽҴҬBҮҺBҼBһBҮҽҰҲҷPҶ ҲҬҲӀFKFҮҷBҴPҶKFҮҲҷҲӀҲұBҶFҺFӓFҮҽҰҲ eϜFҴҵҲһҶPҮBKFҹPҬҺӂҲҷBҼPҭ KF ҮҲҷҲӁҷPҭҴҬBҮҺBҼB

ҲҬҲӀPҶKFҮҲҷҲӁҷFҮҽҰҲҷF e

ϓBҹҺFҶҲҷBҹPҵҲFҮBҺB

"ҴPһҽ T T1 Ҳ T2 ҭFPҶFҼҺҲKһҴBҼFҵBҼBҴҬBҮBKF

ҫFұұBKFҮҷҲӁҴҲҿҽҷҽҼҺBӂӓҲҿҼBӁBҴB PҷҮBKF V (T )= V (T1 )+ V (T2 ).

"ҴPKFҭFPҶFҼҺҲKһҴPҼFҵP T

"ҴPKF E ҴPӀҴBһBҲҬҲӀPҶKFҮҲҷҲӁҷFҮҽҰҲҷF e PҷҮBKF V (E)=1

BKӁFӂӔFҽҶFһҼP V (T ) ҹҲӂFҶPһBҶP V BҴPKFҲұҴPҷҼFҴһҼBKBһҷPPҴPҶһF ҭFPҶFҼҺҲKһҴPҶҼFҵҽҺBҮҲ

0ҮҺFӋҲҬBӔFҶPұBҹҺFҶҲҷFҷFҴҲҿҹPҵҲFҮBҺB ϓBҹҺFҶҲҷBҴҬBҮҺB ҹҺBҬPҽҭҵPҭҹBҺBҵFҵPҹҲҹFҮB

ϞүҸҺүҶҪ ϓBҹҺFҶҲҷBҹҺBҬPҽҭҵPҭҹBҺBҵFҵPҹҲҹFҮBKFҮҷBҴBKFҹҺPҲұҬPҮҽ

ӓFҭPҬFҼҺҲҮҲҶFҷұҲKF

ϐҸҴҪұ ϜBұҶPҼҺҲҶPҹҺҬPһҵҽӁBKҴBҮһҽ ҮҲҶFҷұҲKFҹBҺBҵFҵPҹҲҹFҮB a b Ҳ c ҹҺҲҺPҮ ҷҲڈҺPKFҬҲҝҼPҶһҵҽӁBKҽ һBҹҺBҬBҶBҴPKFһҽ ҹBҺBҵFҵҷFһҼҺBҷҲӀBҶBPһҷPҬF ABCD ҼBPһҷPҬBҶPҰFҮBһFҲұҮFҵҲҷB ab KFҮҲҷҲӁҷҲҿҴҬBҮҺBҼB һҵ "ҴPһFҷBһҬBҴҲPҮ

ҼҲҿҴҬBҮҺBҼBҹPһҼBҬҲKFҮҲҷҲӁҷBҴPӀҴB ҮPڈҲӔFһFһҵPKӁҲKBKFҬҲһҲҷBKFҮҷBҴBKFҮҲҷҲӀҲ ҮҽҰҲҷF һҵ ҠFPҹBҺBҵFҵPҹҲҹFҮҶPҰFһFҹPҹҽҷҲҼҲһB c ҼBҴҬҲҿһҵPKFҬB һҵ ϐB ҴҵF ҹҺBҬPҽҭҵҲҹBҺBҵFҵPҹҲҹFҮKFҹPҹҽӓFҷһB abc ҮҲһKҽҷҴҼҷҲҿKFҮҲҷҲӁҷҲҿҴPӀBҴBҹB KFӓFҭPҬBұBҹҺFҶҲҷB V = abc.

ڈҺPKFҬҲҶB ҼBҮBһFһҬPӋFӓFҶҼҲҿڈҺPKFҬBҷBұBKFҮҷҲӁҴҲҲҶFҷҲҵBӀ

ұBҹҺFҶҲҷBһҬBҴFPҮӓҲҿKF

ҬӒBҶPұڈPҭһҵPҰFҷPһҼҲ

ұҬPҮ

B a =4 DN b =5 DN c =8 DN V = abc =4 5 8=160 DN3 ڈ V =560 N3 b =8 N c =7 N a = V bc = 560 8 · 7 =10 N

Ҭ V =270 EN3 a =5 EN c =9 EN b = V ab = 270 5 · 9 =6 EN

FҴBKF

ϓBҲұҺBӁҽҷBҬBӓFұBҹҺFҶҲҷBҭFPҶFҼҺҲKһҴҲҿҼFҵBӁFһҼPһFҴPҺҲһҼҲҹҺҲҷӀҲҹ ҴPKҲKFҾҸҺҶҽҵҲһBPҲҼBҵҲKBҷһҴҲҶBҼFҶBҼҲӁBҺ BҵҲҵFKFҬҽӁFҷҲҴ ύPҷBҬFҷҼҽҺB ϖBҬBҵҲFҺҲ #POBWFOUVSB$BWBMJFSJ o

ϖBҬBҵҲFҺҲKFҬҹҺҲҷӀҲҹ "ҴPһFҮҬBҼFҵBҶPҭҽҮPҬFһҼҲҽҼBҴBҬҹPҵPҰBKҮBҲҿһҬBҴBҺBҬBҷҴPKB ҲҿһFӁF BҹBҺBҵFҵҷBKFҮBҼPKҺBҬҷҲ һFӁFҹPҹҺFһFӀҲҶBKFҮҷBҴҲҿҹPҬҺӂҲҷB PҷҮBҼBҮҬBҼFҵBҲҶBKҽKFҮҷBҴFұBҹҺFҶҲҷF

ϖBҬBҵҲFҺҲKFҬҹҺҲҷӀҲҹҹҺҲҿҬBҼBҶPҫFұҮPҴBұB KFҺҮPҴBұұBҿҼFҬBұҷBӓFҬҲ ӂFҶBҼFҶBҼҲҴF5BKҹҺҲҷӀҲҹӔFҶPҴPҺҲһҼҲҼҲҽҮBӒFҶҲұҵBҭBӓҽKFҺPҶPҭҽӔBҬB ҮFҮҽҴҼҲҬҷPҮPҴBұҲҬBӓFҶҷPҭҲҿҼүҸҺүҶBҴPKFһFPҮҷPһFҷBұBҹҺFҶҲҷFҭFPҶF

PһҷPҬBһҽ KFҮҷBҴF ҼK BABCDE = BMNPQ ҮBҼP ҼPһҽ ҲҹPҬҺӂҲҷFҹҺFһFҴBһҪҺҪҬҷҲ γ KFҮҷBҴF ҼK BA2 B2 ...E2 = BM2 N2 P2 Q2 ϐBҴҵF ҹҺFһFӀҲҹҺҲұҶFҲҹҺBҬPҽҭҵPҭҹBҺBҵFҵP ҹҲҹFҮBڈҲҵPҴPKPҶҺBҬҷҲ γ ҴPKBKFҹBҺBҵFҵҷBҺBҬҷҲPһҷPҬB ҲҶBKҽ KFҮҷBҴFҹPҬҺӂҲҷF BPһҷPҬҽ ϖBҬBҵҲFҺҲKFҬPҭҹҺҲҷӀҲҹBҼBҮҬBҼFҵBҲҶBKҽ KFҮҷBҴFұBҹҺFҶҲҷF ҼK

ҞҸҺҶҽҵҲӂҲҶPҹҺҬPKFҮҷҽҹPҶPӔҷҽҼүҸҺүҶҽ ϞүҸҺүҶҪ ϐҬFҹҲҺBҶҲҮFһBPһҷPҬBҶBKFҮҷBҴҲҿҹPҬҺӂҲҷBҲKFҮҷBҴҲҶҬҲ һҲҷBҶBҲҶBKҽ KFҮҷBҴFұBҹҺFҶҲҷF

5FPҺFҶBһFҵBҴPҮPҴBұҽKFҹҺҲҶFҷPҶһҬPKһҼBҬBҹBҺBҵFҵҷPҭҹҺFһFҴBҹҲҺBҶҲ ҮFҲϖBҬBҵҲFҺҲKFҬPҭҹҺҲҷӀҲҹB

ϞүҸҺүҶҪ

PһҷPҬFҲҬҲһҲҷF

ϜBһҼPKBӓFPҮҼBӁҴF V ҮPҺBҬҷҲҼҺPҽҭҵB ACA1 KFҮҷBҴPҺBһҼPKBӓҽ PҮҼBӁҴF V ҮP

ҺBҬҷҲҼҺPҽҭҵB CC1A1

ϔһҼPҼBҴPKF VVABC = VCVA1 C1 ,

KFҺKF

PΔABC = PΔVA1 C1

ҺBһҼPKBӓFPҮҼBӁҴF V ҮPҺBҬҷҲҼҺPҽҭҵB ABC KFҮҷBҴPҺBһҼPKBӓҽ PҮҼBӁҴF C ҮP

ҺBҬҷҲҼҺPҽҭҵB A1VC1

ϔұ Ҳ һҵFҮҲKFҮҷBҴPһҼұBҹҺFҶҲҷBһҬFҼҺҲҹҲҺBҶҲҮF

VVABC = VVCA1 C1 = VVACA1

ϐBҴҵF ҹҺҲұҶBKFҺBұҵPҰFҷBҷBҼҺҲҹҲҺBҶҲҮFKFҮҷBҴҲҿұBҹҺFҶҲҷB ҹBKFұBҹҺF ҶҲҷBһҬBҴFPҮҼҲҿҹҲҺBҶҲҮBKFҮҷBҴBҼҺFӔҲҷҲұBҹҺFҶҲҷFҹҺҲұҶF ҼK 1 3 BH BPһҷPҬҽ ҼүҸҺүҶFұBҴӒҽӁҽKFҶPҮBұBҹҺFҶҲҷBҹҲҺBҶҲҮFҷFұBҬҲһҲPҮPڈҵҲҴBPһҷPҬF ҷFҭPһB ҶPPҮҹPҬҺӂҲҷFPһҷPҬFҲҬҲһҲҷF ҺFҶBҼPҶF ұBҹҺFҶҲҷBڈҲҵPҴPKFҹҲҺBҶҲҮFKFҮҷBҴB KFҼҺFӔҲҷҲұBҹҺFҶҲҷFҹҺҲұҶFҴPKBҲҶBһBҼPҶҹҺҲұҶPҶKFҮҷBҴ

ϓBҹҺFҶҲҷBҹPҵҲFҮBҺB

Ҭ V =150 EN3 B =50 EN2 H = 3V B = 3 · 150 50 =9 EN

ҺҲҶүҺ ώҲһҲҷBҹҺBҬҲҵҷFӁFҼҬPҺPһҼҺBҷF ҴҬBҮҺBҼҷF ҹҲҺBҶҲҮFKF H =12 DN BұBҹҺFҶҲҷB V =400 DN3 0ҮҺFҮҲҶPҮ

Ϝүӂүӓү PҬҺӂҲҷBPһҷPҬFKF B = 3V H = 3 · 400 12 =100 DN 2 .

ϖBҴPKFPһҷPҬBҴҬBҮҺBҼҹPҬҺӂҲҷFDN2 ҼPKFһҼҺBҷҲӀBҼPҭBҴҬBҮҺBҼB PһҷPҬҷBҲҬҲ ӀB a = √100=10 DN ϓBҹҺFҶҲҷBұBҺҽҫӒFҷFҹҲҺBҶҲҮF ϞүҸҺүҶҪ

V = H 3 (B + BB1 + B1 ) ϐҸҴҪұ ϐPҹҽҷҲҶPұBҺҽڈӒFҷҽ ҹҲҺBҶҲҮҽ ҮPҹҽҷFҹҲҺBҶҲҮF һҵҲҴB ϐPڈҲKFҷB ҹҲҺBҶҲҮBҲҶBPһҷPҬ

ҺҲҶүҺ ϓBҹҺFҶҲҷBұBҺ

ҬҲһҲҷB H =48 DN BұڈҲҺҹPҬҺӂҲҷBPһҷPҬBKF B + B1 =164 DN 2 0ҮҺFҮҲҼҲҹPҬҺӂҲҷ

ҺBҶҲҮF

Ϝүӂүӓү ϖBҴPKF H =48 DN ҲһPڈұҲҺPҶҷBҾҸҺҶҽҵҽ ұBұBҹҺFҶҲҷҽ ұBҺҽڈӒFҷF

ҹҲҺBҶҲҮF ҲҶBҶPһҲһҼFҶKFҮҷBӁҲҷB 16(B + BB1 + B1 )=3904, B + B1 =164,

BB1 =80 Ҽӑ BB1 =6400.

ҽ Ҳ B Ҳ B1 һҽ ҺFӂFӓBҴҬBҮҺBҼҷFKFҮҷBӁҲҷF x 2 164x +6400=0. ϐBҴҵF B = x1 =100 DN 2 B1 = x2 =64 DN 2

ϓBҹҺFҶҲҷBҹPҵҲFҮBҺB

ϓ"ϐ"Ҡϔ

ϔҬҲӀBKFҮҷFҴPӀҴFKFҹҽҼBҶBӓBPҮҲҬҲӀFҮҺҽҭF0ҮҺFҮҲPҮҷPһұBҹҺFҶҲҷBҼF ҮҬFҴPӀҴF

ϖPҵҲҴPҹҽҼBKFҲҬҲӀBKFҮҷFҴPӀҴFҬFӔBPҮҲҬҲӀFҮҺҽҭFҴPӀҴFBҴPKFұBҹҺFҶҲҷB ҹҺҬFҴPӀҴFҮҬBҹҽҼBҬFӔBPҮұBҹҺFҶҲҷFҮҺҽҭF

ϓBҹҺFҶҲҷBҴPӀҴFKFDN3 0ҮҺFҮҲӓFҷҽҹPҬҺӂҲҷҽ

PҬҺӂҲҷBҴPӀҴFKFDN2 0ҮҺFҮҲҮҽҰҲҷFҲҬҲӀFҲҮҲKBҭPҷBҵFҲұBҹҺFҶҲҷҽ

ҴPӀҴF

ϐBҼFһҽҼҺҲҶFҼBҵҷFҴPӀҴFһBҲҬҲӀBҶBҮҽҰҲҷFDN DNҲDN .PҰFҵҲһFPҮ ҼFҼҺҲҴPӀҴFҲұҵҲҼҲKFҮҷBҴPӀҴBһBҲҬҲӀPҶҮҽҰҲҷFDN

ϐBҵҲҫҲҶPҭBPҮBҹPҮҲҭҷFӂҴPӀҴҽPҮұҵBҼBһBҲҬҲӀPҶҮҽҰҲҷFEN ϏҽһҼҲҷB ұҵBҼBKF γ ≈ 19, 3 HDN3

0һҷPҬBҹҺBҬPҭҹBҺBҵFҵPҹҲҹFҮBKFҹBҺBҵFҵPҭҺBҶһBһҼҺBҷҲӀBҶBҮҽҰҲҷFDNҲ DNҲҮҲKBҭPҷBҵPҶҮҽҰҲҷFDN PҬҺӂҲҷBҹBҺBҵFҵPҹҲҹFҮBKFDN2 0ҮҺF

ҮҲӓFҭPҬҽұBҹҺFҶҲҷҽ

ϚһҷҸҬҪҹҺҪҬҸҭҹҪҺҪҵүҵҸҹҲҹүҮҪӑүҺҸҶҫӁҲӑҪӑүҹҸҬҺӂҲҷҪN2 ҸҬҺӂҲҷүҮҲӑҪ ҭҸҷҪҵҷҲҿҹҺүһүҴҪһҽN2 ҲN2 ϔұҺҪӁҽҷҪӑұҪҹҺүҶҲҷҽҹҪҺҪҵүҵҸҹҲҹүҮҪ ύҪұүҷӑүҮҽҭN ӂҲҺҸҴN ҪҮҷҸKFҷҪҭҷҽҼҸ

ϚһҷҸҬҷҪҲҬҲӀҪҹҺҪҬҲҵҷүӁүҼҬҸҺҸһҼҺҪҷүҹҲҺҪҶҲҮүҲҶҪҮҽҰҲҷҽ

ϖүҸҹһҸҬҪҹҲҺҪҶҲҮҪҲҶҪҬҲһҲҷҽNҲҴҬҪҮҺҪҼҷҽҸһҷҸҬҽһҪҲҬҲӀҸҶҮҽҰҲҷү NҟҸҮҷҲӀҲҲҮҬҸҺҪҷүұҪҽұҲҶҪӑҽӓүҷүұҪҹҺүҶҲҷүϖҸҵҲҴҸҫҲҴҪҶҲҸҷҪ ҷҸһҲҬҸһҼҲUҫҲҵҸҹҸҼҺүҫҷҸҮҪҹҺүҬүұүҴҪҶүҷҸҮҴҸҭҪӑүҲұҭҺҪӋүҷҪ ϏҽһҼҲҷҪҴҪ ҶүҷҪӑү ρ =2, 5HDN3

ϚһҷҸҬҷҪҲҬҲӀҪҹҺҪҬҲҵҷүӁүҼҬҸҺҸһҼҺҪҷүҹҲҺҪҶҲҮүҲҶҪҮҽҰҲҷҽ a ҮҸҴӑүҽҭҪҸҲұ ҶүӋҽҮҬүһҽһүҮҷүҫҸӁҷүҲҬҲӀү 60◦ ϚҮҺүҮҲұҪҹҺүҶҲҷҽҹҲҺҪҶҲҮү

3 < 3, 32211 ...

10π =1385, 45 ...,

π =3, 14159265 ...

ҸұҷҪӁүҷҲҿҷҪһҵҲҴҪҶҪҲ

ҺҪҾҲҴҷҪҶҹҺүҮҸӁҪҬҪӑҸӂҷүҴҪһҬҸӑһҼҬҪҾҽҷҴӀҲӑү

y = f (x) Ҳ y = f (x

f1(x)=5x f2(x)= x 5 f3(x)= x 5x f4(x)=5+5x f5(x)= √5x f6(x)=(√3) x f7 = {(x,π x ) | x ∈ R}

ϚҮҺүҮҲ f (5) f ( 3) f ( x) f (f (0)) ҪҴҸӑү Ҫ f (x)=7x ҫ f (x)=5 x Ҭ f (x)=(0, 2)2x

ϖҸӑүһҽҸҮһҵүҮүӔҲҿҾҽҷҴӀҲӑҪҺҪһҼҽӔү ҪҴҸӑүҸҹҪҮҪӑҽӔү

Ҭ f (x)= 2 3 x Ҳ g(x)= 3 2 x Ҫ

Ҳ f3(x)= 2x

f4(x)=2x+2

f2(x)=2x +5

f (x)= 3 2 x 2ҫ f (x)=1 2x Ҭ f (x)=2 3x +0, 5

f (x)=73x ҫ f (x)=51 2x Ҭ f (x)=3 5x 1ҭ f (x)=5 103x+5 11

5m > 5nҫ

όҴҸӑү a ∈ R+ ҽҷүһҲҸұҷҪҴҽҺүҵҪӀҲӑү

ϚҹӂҼҲҸҫҵҲҴҸҬҪҴҬҲҿүҴһҹҸҷүҷӀҲӑҪҵҷҲҿӑүҮҷҪӁҲҷҪӑү

Ҫ MPH10 100=2 ӑүҺӑү 102 =100

2 256=8 ӑүҺӑү 28 =256 Ҭ MPH3 1 27 = 3

* aMPHa b = b ϚҬҪӑүҮҷҪҴҸһҼӑүҸӁҲҭҵүҮҷҪҵҸҭҪҺҲҼҪҶӑүҽҹҺҪҬҸҷҪҬүҮүҷҲүҴһҹҸҷүҷҼ

a ac = c

Ҫ MPH7 72 =2ڈ MPH0,1 0, 1√2 = √2ϚڈӑҪһҷҲҼҲ

ϐҸҴҪҰҲҼүһҪҶҲҸҬҸһҬҸӑһҼҬҸҹҸҽұҸҺҽҷҪҮҸҴҪұһҬҸӑһҼҬҪ*7

Ҫ MPH10 x 100 = MPH10 x MPH10 100= MPH10 x 2

ڈ MPHa π 2 = MPHa π MPHa 2 7* MPHa br = r MPHa b (b> 0)

a 0, 0081=4Ү MPHa 4, 84=2

Ҫ MPHa (3 7x)ҫ MPHa (x 2 + x 6)Ҭ MPHa 8 x 3+ x

ҫ 3 MPHx a 5 MPHx b 7 MPHx c

Ҭ MPHa (p + q)+ MPHa (p q) (MPHa p + MPHa q)

ҭ MPHa 7+ MPHa √5 1 2 MPHa 113 Ү 1 2 MPHa (x + y) 2 3 (MPHa x + MPHa y)

Ӌ MPHx a + 1 3 MPHx b + 1 4 MPHx c + 1 5 MPHx (d + e)

MPHa x = MPHb x MPHb a

a b = 1 MPHb a a =1 b =1

a b = MPH 1 a b

MPHx (5x 2 ) MPH5 x =3

f 1 (4)=2 f 1 (8)=3

ҼҺҲӁҷҲһҽҽҸҮҷҸһҽҷҪҹҺҪҬҽ y = x һҵ

ϚҫҺҪұҵҸҰҲҶҸҸҬҸҷҪҹҺҲҶү

ҺҽҵҲҷүҪҺҷүҾҽҷҴӀҲӑү y = ax+b Ҳ

ӓҸӑҲҷҬүҺұҷү y = 1 a x b a όҴҸҼҪӁ

ҴҪ A(m,n) ҹҺҲҹҪҮҪҭҺҪҾҲҴҽҹҺҬү

m = an + b

Ҫ f (x)=2xҫ f (x)= 3xҬ f (x)=0, 2x +0, 3 ҭ f (x)=1 2xҮ

f (x) Ҳ f 1 (x) ҪҴҸӑү

Ҫ f (x)=3x +1; ҫ f (x)= 2x +5Ҭ f (x)=0, 7x

ϚҮҺүҮҲҲҷҬүҺұҷҽҾҽҷҴӀҲӑҽҲҸҫҺҪұҵҸҰҲҺүұҽҵҼҪҼҭҺҪҾҲӁҴҲҪҴҸӑү Ҫ f (x)= xҫ f (x)= x +1Ҭ f (x)= x

A(m,n)

f (x)

ϖҪҴҪҬӑүҸҮҭҸҬҸҺҷҪҹҲҼҪӓүҲұҹҺүҼҿҸҮҷҸҭұҪҮҪҼҴҪҪҴҸһүҮҸҮҪҽһҵҸҬ

(x)= MPHa x,a> 0,a =1

ҮҺҽҭҲҿҾҽҷҴӀҲӑҪҲһҹҲҼҲҬҪҵҲ

ҾҽҷҴӀҲӑҪMPHa x ҲMPHa ( x) ҸҭҪҺҲҼҪҶһҴҪҾҽҷҴӀҲӑҪӑүҫҲӑүҴӀҲӑҪ

B ҲұMPHa x1 = MPHa x2

ҪҬүҮҲҼүһҪҶҲҽһҵҸҬүҹҸҮҴҸӑҲҶҪҺүӂүӓүҲҶҪһҶҲһҵҪ ϓόϐόҠϔ ҝҲһҼҸҶҴҸҸҺҮҲҷҪҼҷҸҶһҲһҼүҶҽһҴҲӀҲҺҪӑҾҽҷҴӀҲӑү f (x) Ҳ g(x) ҪҴҸӑү

Ҫ f (x)= 4 3 x Ҳ g(x)= MPH 3 4 x

ҫ f (x)=4x Ҳ g(x)= MPH4 x

Ҭ f (x)=0, 3x Ҳ g(x)= MPH0,3 x

ҭ f (x)= 1 10 x Ҳ g(x)= MPH 1 10 x

Ү f (x)= 3 5 x

Ҫ f (x)= MPHa (3x 7)ҫ f (x)= MPHa ( 7x)

Ҭ f (x)= MPHa (3 5x)ҭ f (x)= MPHa ((5x 3)(2 x))

Ү f (x)= MPHa 3 x 2x 7 Ӌ f (x)= MPHa (x 2 9)

f (x)=1+ MPH3 xҫ f (x)=2 MPH0,3 xҬ f (x)=3 MPH2 x ҭ f (x)= 5+ MPH10 xҮ f (x)= MPH0,6 ( x)Ӌ f (x)= MPH2 (x +4)

ϖҸӑҪҸҮһҵүҮүӔҲҿҾҽҷҴӀҲӑҪҺҪһҼү ҪҴҸӑҪҸҹҪҮҪҽӁҲҼҪҬҸҶҮҸҶүҷҽ

f1 (x)= MPHa ( x) a> 1 f2 (x)= MPHa x a> 1 f3 (x)= MPHa ( x) 0 <a< 1 f4 (x)= MPHa x 0 <a< 1 f5 (x)= MPHa ( x) a> 1 f6 (x)= MPHa ( x) 0 <a< 1

ϓҪҴҸӑҽҬҺүҮҷҸһҼҪҺҭҽҶүҷҼҪ x ӑү f (x) > 0 ҸҮҷҸһҷҸ f (x) < 0 ҪҴҸӑү

(

Ҫ MPH7 2, 5 · MPH2,5 7 > 0ҫ

ϜүӂҲӑүҮҷҪӁҲҷү ҹҺүҼҿҸҮҷҸҷҪҬүҮҲҽһҵҸҬүҹҸҮҴҸӑҲҶҪҸҷүҲҶҪӑҽһҶҲһҵҪ

Ҫ MPH x =2 MPH 4+ 1 3 MPH 27 1 2 MPH 64

ҫ MPH 2= MPH(x 5) 3 MPH 3

Ҭ MPH(x 3) MPH 6=2 MPH 3+ MPH(x +2)

ҭ MPH 3+ 1 2 MPH 4+ MPH(5x 1)= MPH(x +2)+ MPH 23

ϚҫӑҪһҷҲұҪӂҼҸһҽҽҷҪҬүҮүҷҲҶӑүҮҷҪӁҲҷҪҶҪҲұҸһҼҪҬӒүҷүҸһҷҸҬү

ϚҬҪӑҮҸҴҪұһҹҺҸҬүҮүҷӑүҵҸҭҲӁҴҲҶҹҸһҼҽҹҴҸҶҴҸӑҲһүұҸҬү

ҲҸҹҲһҽӑүһүҼҪҽҼҸҵҸҭҲӑҸҶ

MPH r = k + MPH p

ҽ ҹҸһҵүҮӓҲҿҺҪұҶҪҼҺҪӓҪӑүMPH 28= MPH 2, 8 10= MPH 2, 8+ MPH 10=(MPH 2, 8)+1=1+ MPH 2, 8

MPH 156, 7=2+ MPH 1, 567

MPH 0, 0013= 3+ MPH 1, 3

MPH 9978, 256=3+ MPH 9, 978256

MPH 0, 5= 1+ MPH 5.

k + MPH p

MPH 0, 05= 2+ MPH 5,

, 32,

0, 0314=3, 14 · 10 2 ; MPH 0, 0314= 2+ MPH 3, 14

0, 314=3, 14 · 10 1 ; MPH 0, 314= 1+ MPH 3, 14 3, 14=3, 14 · 100 ; MPH 3, 14=0+ MPH 3, 14 31, 4=3, 14 · 101 ; MPH 31, 4=1+ MPH 3, 14

314=3, 14 102 ; MPH 314=2+ MPH 3, 14

3140=3, 14 103 ; MPH 3140=3+ MPH 3, 14 ҲҼҮ

MPH

ҸҮҷҽҵү

ҺҲҶүҺ

MPH 0, 00075= 4+ MPH 7, 5; ҴҪҺҪҴҼүҺҲһҼҲҴҪ 4

MPH 0, 2= 1+ MPH 2; ҴҪҺҪҴүҺҲһҼҲҴҪ 1

MPH 0, 055= 2+ MPH 5, 5;

2 ҲҼҮ

0, 305MPH 80, 07 MPH 506MPH 123 MPH 0, 0001010MPH 5505MPH 123, 4MPH 1

MPH 917, 4=2, 96256

MPH 917, 4=2,

0, 02449

MPH 0, 017905=0, 25297 2= 1, 74703

MPH 0, 017905= 1, 7470257

103,02458

ҽ ҼҪһҼүҺҪ 10x ұҪ x =3, 02458

MPH x = 3, 32982.

ҺҲҫҵҲҰҷҪҬҺүҮҷҸһҼҺүӂүӓҪ ҽұҹҸҶҸӔҴҪҵҴҽҵҪҼҸҺҪ ӑүҫҺҸӑ 10 3,32982 ҸҮҷҸ һҷҸ x =0, 0004679290

ҝұҹҸҶҸӔҵҸҭҪҺҲҼҪҶһҴҲҿҼҪҫҵҲӀҪҺҪҮҲҶҸҸҬҪҴҸ MPH x = 3, 32982+4 4

ϔұҺҪӁҽҷҪӑҫүұҴҸҺҲӂӔүӓҪҼҪҫҵҲӀҪҲҵҲҴҪҵҴҽҵҪҼҸҺҪһҵүҮүӔүҮүҴҪҮҷүҵҸҭҪҺҲҼҶү

Ҫ MPH 3 √10 ҫ MPH 1 100

Ҭ MPH 1000√2 ҭ MPH(100)MPH √10 100

ҪҹҲӂҲҽҸҫҵҲҴҽ k + MPH p k ∈ Z 1 p< 10 һҵүҮүӔүҵҸҭҪҺҲҼҶү

Ҫ MPH 278, 56ҫ MPH 0, 0505Ҭ MPH 2, 708ҭ MPH 3333

Ү MPH 0, 00008Ӌ MPH 0, 12345ү MPH 40 Ұ MPH 53, 2 102

ϖҪҮҪӑүMPH 7, 614=0, 88161 ҸҮҺүҮҲ MPH 7614MPH 761, 4MPH 76, 14MPH 0, 7614MPH 761400MPH 0, 0007614

όҴҸһҽһҵүҮүӔҲҫҺҸӑүҬҲҬҺүҮҷҸһҼҲҮүҴҪҮҷҲҿҵҸҭҪҺҲҼҪҶҪ ҷҪҹҲӂҲҲҿҽҸҫҵҲҴҽ ұҫҲҺҪҷүҭҪҼҲҬҷүҴҪҺҪҴҼүҺҲһҼҲҴүҲҹҸұҲҼҲҬҷүҶҪҷҼҲһү

Ҫ 3, 72546ҫ 2, 31017Ҭ 4, 01239

ҭ 1, 73220Ү 0, 25032Ӌ 0, 01101

2=0, 30103 MPH 3=0, 47712 MPH 7=0, 84510

Ҫ 3, 45128ҫ 0, 23550 1Ҭ 0, 27715

ҭ 3, 14145Ү 1, 33491Ӌ 0, 25098 3

Ҫ 92, 76 0, 9276 9,

Ҭ 1123, 8 112370 0, 11237 112379

ҭ 0, 0008 0, 70007 6, 0066 1, 0101

ϚҮҺүҮҲҫҺҸӑӁҲӑҲӑүҮүҴҪҮҷҲҵҸҭҪҺҲҼҪҶ

Ҫ 3, 30103 0, 047712 2, 15126 4, 77112 0, 00007

ҫ 0, 35411 2 1+0, 43905 0, 01901 4

Ҭ 2, 11905 3, 12105 2, 55407 0, 07708

(π =3, 14159265 ...)

a =13, 70,4

MPH a =0, 4 · MPH 13, 7

a =0, 4 · 1, 13672,

a =0, 45688

a =100,454688 .

x =1, 5849625.)

ϐҪҫҲһҶҸҹҺҲҶүҷҲҵҲҵҸҭҪҺҲҼҪҶһҴүҼҪҫҵҲӀү ҵҸҭҪҺҲҼҶҽӑҶҸҾҸҺҶҽҵҽ

MPH x = MPH 47712 MPH 30103, MPH x =4, 67863 4, 47861, MPH x =0, 20002.

ϚҮҪҬҮүӑү x =1, 5849

όҴҸӑүҹҸҼҺүҫҷҸҺҪӁҽҷҪҼҲһҪҷүҭҪҼҲҬҷҲҶҫҺҸӑүҬҲҶҪ ҪұҪӓҲҿ ҴҪҸӂҼҸһү ұҷҪ ҵҸҭҪҺҲҼҪҶҷҲӑүҮүҾҲҷҲһҪҷ һҷҪҵҪұҲҶҸһүҷҪһҵүҮүӔҲҷҪӁҲҷ

ҪӋҲҬҺүҮҷҸһҼҲұҺҪұҪ

172, 45 DN 2

s =13, 156 DN

V =0, 5 EN3 ϔҺҪӀҲҸҷҪҵҪҷҫҺҸӑ e =2, 71828

1+ 1 n n ,n =1, 2, 3,...

ϚҮҺүҮҲҹҺҲҫҵҲҰҷүҬҺүҮҷҸһҼҲҫҺҸӑҪ e ҲұҸҬүҾҸҺҶҽҵү ҽұҲҶҪӑҽӔҲұҪ n ҺүҮҸҶҫҺҸ ӑүҬү

όҴҸһүһҪMO x ҸұҷҪӁҲMPHe x ҲұҺҪӁҽҷҪӑ

Ҫ MO eҫ MO 2Ҭ MO 10ҭ MO πҮ MO 1000 ұҪҫҺҸӑ e ҽұҶҲҬҺүҮҷҸһҼ 2, 718 ҪұҪ π

0ύϜ5 "5& "

"ԵጥᡒԱFᡒጥԼጧጥጤԲBԻጥoᡒPԶԸጤBᡓጥԶBԲF ᡕFPԱFᡒԵጥKԶጧFԹጥᡕԸԵጢBᡕFPԱFᡒԵጥKԶጧF ԹጥᡕԸԵጢoᡒPԶԸԲBԻԵᡒBԲFԹԳԵԱԸጨF BጙጥᡔጆጥጨᡖFԵᡒ o

PҬҺӂҴPKBһFҮPҫҲKBPҫҺҼBӓFҶҵҲҷҲKF l PҴPҽҼҬҺӋFҷFҹҺBҬF AB ұBҹҽҷ ҽҭBPҷBұҲҬBһFPҫҺҼҷBҹPҬҺӂ

ҬBҴBҼBӁҴBҵҲҷҲKF l ҹҺҲPҫҺҼBӓҽPҹҲһҽKFҴҺҽҰҷҲӀҽһBӀFҷҼҺPҶҷBPһҲ

AB ҹҺҲӁFҶҽKFҺBҬBҷҴҺҽҰҷҲӀFҷPҺҶBҵҷBҷBPһҽ0ҫҺҷҽҼP BҴPһFPҫҺҼҷBҹP ҬҺӂһFӁFҺBҬҷҲҶBҷPҺҶBҵҷҲҶҷBPһҽ ҽҹҺFһFҴҽһFҮPҫҲKBKҽҴҺҽҰҷҲӀF

ϐүҾҲҷҲӀҲӑҪ

0ҫҺҼҷBҹPҬҺӂҴPKBһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬFҴPKBKFҹBҺBҵFҵҷBPһҲ AB ҷBұҲҬBһFҹҺBҬBӀҲҵҲҷҮҺҲӁҷBҹPҬҺӂ һҵ

ϐүҾҲҷҲӀҲӑҪ

0ҫҺҼҷBҹPҬҺӂҴPKBһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬFҴPKBһFӁFPһҽ BҷҲKFҷPҺҶBҵ ҷBҷBӓҽ ҷBұҲҬBһFҹҺBҬBҴPҷҽһҷBҹPҬҺӂ

5BӁҴBҹҺFһFҴBҲұҹҺFҼҿPҮҷFҮFҾҲҷҲӀҲKFKFҬҺҿҴPҷҽһҷFҹPҬҺӂҲ

ҹPҵҽ ҹҺFӁҷҲҴһҾFҺF

ҬFҼBӁҴFҹPҵҽҴҺҽҰҷҲӀFҹPҮKFҮҷBҴPһҽҽҮBӒFҷFPҮӀFҷҼҺB O ҹBҼPҬBҰҲҲ ұBһҬFҼBӁҴFһҾFҺFϐBҴҵF һҾFҺBKFһҴҽҹһҬҲҿҼBӁBҴBҽҹҺPһҼPҺҽҴPKFһҽҷBҮBҼPҶ ҺBһҼPKBӓҽPҮKFҮҷFҼBӁҴFoӀFҷҼҺBһҾFҺF

FPҶFҼҺҲKһҴPҼFҵPPҭҺBҷҲӁFҷPҹҺBҬPҶӀҲҵҲҷҮҺҲӁҷPҶҹPҬҺӂҲҲҮҬFҶB ҺBҬҷҲҶBҷPҺҶBҵҷҲҶҷBPһҽҼFҹPҬҺӂҲҷBұҲҬBһFҹҺBҬҬBӒBҴ һҵB

ϐFҵPҬҲҹҺFһFӁFҷҲҿҺBҬҷҲPҭҺBҷҲӁFҷҲӀҲҵҲҷҮҺҲӁҷPҶҹPҬҺӂҲҷBұҲҬBKҽһF PһҷPҬFҬBӒҴBϐFPӀҲҵҲҷҮҺҲӁҷFҹPҬҺӂҲҲұҶFӋҽҺBҬҷҲPһҷPҬBKF PҶPҼBӁҬBӒ ҴB ϜBһҼPKBӓFҲұҶFӋҽҮҬFҺBҬҷҲҴPKFһBҮҺҰFPһҷPҬFKF ҬҲһҲҷBҬBӒҴB ϐFҵPҬҲҲұҬPҮҷҲӀBӀҲҵҲҷҮҺҲӁҷFҹPҬҺӂҲҲұҶFӋҽPһҷPҬBһҽ ҲұҬPҮҷҲӀFҬBӒ ҴB 0һҷPҬBҹҺBҬPҭҬBӒҴBKFҴҺҽҭ ϓBӂҼP ҺBҬBҴҺPұӀFҷҼҺFPһҷPҬBҹҺBҬPҭ

c ҲҶBKҽұBKFҮҷҲӁҴҽҼBӁҴҽBҹBҺBҵFҵҷFһҽҮBҼPKҹҺBҬҲ

FӁFӓFҶҴPһFӀҲҵҲҷҮҺҲӁҷFҹPҬҺӂҲҮҬFҶBҺBҬҷҲҶBҹBҺBҵFҵҷҲҶҺBҬҷҲҬPҮҲӒF ҮPҫҲKBһF ҴPһҲҬBӒBҴ һҵҫ

0һҷPҬF PҶPҼBӁ ҲұҬPҮҷҲӀFҲҬҲһҲҷBҴPһPҭҬBӒҴBҮFҾҲҷҲӂҽһFҷBҲһҼҲҷBӁҲҷҴBP ҲҴPҮҹҺBҬPҭҬBӒҴB

FPҶFҼҺҲKһҴPҼFҵPPҭҺBҷҲӁFҷPҹҺBҬPҶҴPҷҽһҷPҶҹPҬҺӂҲҲKFҮҷPҶҺBҬҷҲ

FӁFӓFҶҴPһFҴPҷҽһҷFҹPҬҺӂҲҷFҴPҶҺBҬҷҲҴPKBKFҹBҺBҵFҵҷBҺBҬҷҲҬPҮҲӒFҲҷF ҹҺPҵBұҲҴҺPұҬҺҿ V ҮPҫҲKBһFҴPһBҴҽҹB һҵҫ

0һҷPҬB PҶPҼBӁ ҲұҬPҮҷҲӀBҲҬҲһҲҷBҴPһFҴҽҹFҮFҾҲҷҲӂҽһFҷBҲһҼҲҷBӁҲҷҴBPҲ ҴPҮҹҺBҬFҴҽҹF һҵ

FҴBKF β ҺBҬBҷҴPKBҹҺFһFӀBҴҽҹҽҲҹBҺBҵFҵҷBKFӓFҷPKPһҷPҬҲ FPҶFҼҺҲ KһҴPҼFҵPҴPKFKFҮFPҴҽҹFҲұҶFӋҽӓFҷFPһҷPҬFҲҺBҬҷҲ β ҷBұҲҬBһFұBҺҽ ҫӒFҷBҴҽҹB һҵ

BҺBҵFҵҷҲҹҺFһFҴһBҺBҬҷҲ

ҴPҷҽһҷFҹPҬҺӂҲҲұҶFӋҽPһҷPҬBKF PҶPҼBӁ ұBҺҽҫӒFҷFҴҽҹF

ϓBҺҽҫӒFҷBҴҽҹBҴPKBһFҷBPҹҲһBҷҲҷBӁҲҷҮPҫҲKBҲұҹҺBҬFҴҽҹFҷBұҲҬBһF

ҺBҬҬBӒBҴ ҹҺBҬBҴҽҹBҲұBҺҽҫӒFҷBҴҽҹBҶPҭҽҮBһFҹPһҶBҼҺBKҽҲҴBP PҫҺҼ ҷBҼFҵB

ώBӒBҴһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬPҽҭBPҷҲҴBPҴPKFҮҷFӓFҭPҬFһҼҺBҷҲӀF һҵ

ϖҽҹBһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬPҽҭҵPҭҼҺPҽҭҵBPҴPKFҮҷFӓFҭPҬFҴBҼFҼF һҵ ҵ ҵ

ϓBҺҽҫӒFҷBҴҽҹBһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬPҽҭҵPҭҼҺBҹFұBPҴPһҼҺBҷҲӀFҷB ҴPKҽҷBҵFҰҽҹҺBҬҲҽҭҵPҬҲ һҵ

ҺFһFҴPҫҺҼҷPҭҼFҵBһBҺBҬҷҲKF

B ҹBҺBҵFҵBҷ oBҴPKFҺBҬBҷҹҺFһFҴBҹBҺBҵFҵҷBPһҷPҬҲ ҫ ҽұҮҽҰBҷ oBҴPҺBҬBҷҹҺFһFҴBһBҮҺҰҲҮҬFҲұҬPҮҷҲӀF ҽһҵҽӁBKҽҴҽҹF ҹҺFһFӁҷBҺBҬBҷҼBҮBһBҮҺҰҲҲҬҺҿҴҽҹF

Ҭ PһҷҲ oBҴPҺBҬBҷҹҺPҵBұҲҴҺPұPһҽҼFҵB

BһҵҲҴBҶB ҲҹҺFҮһҼBҬӒFҷҲһҽҺBұҷҲҹҺFһFӀҲPҫҺҼҷҲҿҼFҵB ҵ

ҺBҬBҴҽҹBKF KFҮҷBҴPһҼҺBҷҲӁҷB BҴPKPKKFPһҷҲҹҺFһFҴKFҮҷBҴPһҼҺBҷҲӁBҷ ҼҺPҽҭBP

ұBҮBҼҴBKF aH =120 Ҳ a 2 + H 2 =289

ϔұPҬҲҿKFҮҷBӁҲҷBҮPڈҲKBҶPҮBKF a 2 +2aH + H 2 =529, Ҽӑ (a + H )2 =529

PҮBҴҵFKF a + H =23.

ϔұKFҮҷBӁҲҷB a + H =23 Ҳ a · H =120 ұBҴӒҽӁҽKFҶPҮBһҽ a Ҳ H ҺFӂFӓBҴҬBҮҺBҼҷF

KFҮҷBӁҲҷF x 2 23x +120=0ϜFӂFӓBҼFKFҮҷBӁҲҷFһҽ x1 =15 Ҳ x2 =8 ҺFҶBҼPҶF ұBҮBҼBҴҲҶBҮҬBҺFӂFӓB a1 =15 DN,H1 =8 DNҲ a2 =8 DN,H2 =15 DN PҵҽҹҺFӁҷҲҴPһҷPҬFKF

ϓ"ϐ"Ҡϔ ҢҼBҹҺFҮһҼBҬӒBһҴҽҹһҬҲҿҼBӁBҴBҽҹҺPһҼPҺҽҹPҮKFҮҷBҴPҽҮBӒFҷҲҿPҮҮBҼFҹҺBҬF "ҴPһFҷBҺBҬBҷ α PһҷPҭҹҺFһFҴBҹҺBҬPҭҬBӒҴBҹPһҼBҬҲҷPҺҶBҵҷBҺBҬBҷ β ҴҺPұKFҮ ҷҽҲұҬPҮҷҲӀҽҴPKBҹҺҲҹBҮBPһҷPҶҹҺFһFҴҽ PҷҮBҺBҬBҷ β ҷFҶBһBҬBӒҴPҶҮҺҽҭҲҿ ұBKFҮҷҲӁҴҲҿҼBӁBҴBPһҲҶҼBӁBҴBҼFҲұҬPҮҷҲӀFϐPҴBҰҲҼP BҹPҶFҷB ϜBҬBҷ β ҷBұҲҬBһF ҼBҷҭFҷҼҷBҺBҬBҷ ҬBӒҴB

ϐPҴBҰҲҮBKFҹPҬҺӂҲҷBPһҷPҭҹҺFһFҴBҹҺBҬPҭҬBӒҴBҬFӔBPҮҹPҬҺӂҲҷFһҬBҴPҭ ҮҺҽҭPҭҽұҮҽҰҷPҭҹҺFһFҴBҼPҭҬBӒҴB

ϐҬBҹҺBҬBҬBӒҴBҲҶBKҽKFҮҷBҴFҬҲһҲҷF H =25 DN PһFһҽҲҶҹBҺBҵFҵҷF BPһҷPҬF ҵFҰFҽҮҬFҶBҺBҬҷҲҶB PҵҽҹҺFӁҷҲӀҲPһҷPҬBһҽҲҶ R =44 DNҲ r =37 DN B ҺBһҼPKBӓFҲұҶFӋҽӀFҷҼBҺBPһҷPҬBҴPKFҵFҰFҽҲһҼPKҺBҬҷҲKF c =15 DNϔұҺBӁҽҷBK ҹPҬҺӂҲҷҽұBKFҮҷҲӁҴPҭҽұҮҽҰҷPҭҹҺFһFҴBҼBҮҬBҬBӒҴB

ϜBҬBҷҹBҺBҵFҵҷBPһҲҹҺBҬPҭҬBӒҴB ҷBҺBһҼPKBӓҽDNPҮPһFPҮһFӀBPҮPһҷPҬF ҬBӒҴBҵҽҴPҮ 120◦ ϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҽұҮҽҰҷPҭҹҺFһFҴBBҴPKFҬҲһҲҷBҬBӒҴB DN

0ҮҷPһҹPҬҺӂҲҷFPһҷPҬFҹҺBҬPҭҬBӒҴBҲҹPҬҺӂҲҷFPһҷPҭҹҺFһFҴBKF π :4

ҽҭBPҲұҶFӋҽҮҲKBҭPҷBҵBPһҷPҭҹҺFһFҴB

ϔұҬPҮҷҲӀBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFҲҶBҮҽҰҲҷҽDNҲһBҺBҬҷҲPһҷPҬFұBҴҵBҹB ҽҭBPPҮ 60◦ PҵҽҹҺFӁҷҲҴKFҮҷFPһҷPҬFKFҮҬBҹҽҼBҬFӔҲPҮҹPҵҽҹҺFӁҷҲҴBҮҺҽҭF 0ҮҺFҮҲPҫBҹPҵҽҹҺFӁҷҲҴB

PҬҺӂҲҷFPһҷPҬBұBҺҽҫӒFҷFҴҽҹFһҽDN2 ҲDN2 χFҷBҬҲһҲҷBҹPҮFӒFҷBKF ҷBҼҺҲҹPҮҽҮBҺҷBҮFҵBҲҴҺPұһҬBҴҽҼBӁҴҽҹPҮFҵFҹPһҼBҬӒFҷBKFҺBҬBҷҹBҺBҵFҵҷB PһҷPҬҲ0ҮҺFҮҲҹPҬҺӂҲҷFҹҺFһFҴB

ϛPKBҶҽҹҲһBҷFҹҺҲұҶFҴPҺҲһҼҲӔFҶPұBҲұҬPӋFӓFҾҸҺҶҽҵFұBҹPҬҺӂҲҷҽ ҬBӒҴBϛҺҲҼPҶFҷFӔFҶPҮBҬBҼҲҹPҼҹҽҷҲһҼҺPҭҮPҴBұKFҺKFұBҼPҷFPҹҿPҮҷP ҹPұҷBҬBӓFһBҮҺҰBKBҶBҼFҶBҼҲҴFҲұҬҲӂҲҿҺBұҺFҮB ҷFҭPӔFҶPҮBҼҲһҴҲӀҽҮP ҴBұBPһҵBӓBKҽӔҲһFҷBҷFҴBҲҷҼҽҲҼҲҬҷBҺBұҶBҼҺBӓB ϛPһҶBҼҺBKҶPҹҺBҬҲҵҷҽҹҺҲұҶҽ

ҹPҵҽҹҺFӁҷҲҴBPһҷPҬF

KFҮBKFPҫҲҶPһҷPҬFҽҹҲһBҷFҹҺҲұҶFҶBӓҲPҮPҫҲҶBPһҷPҬFҬBӒҴB ϓBӂҼP .FӋҽҼҲҶ BҴPҫҺPKһҼҺBҷҲӀBҹҺBҬҲҵҷPҭҶҷPҭPҽҭҵBPһҷPҬFҽҮҬPһҼҺҽӁҲҶP ҼB ҺBұҵҲҴBҲұҶFӋҽPҫҲҶBPһҷPҬFҬBӒҴBҲPҫҲҶBҽҹҲһBҷFҹҺҲұҶFӔFһFһҶBӓҲҼҲ

5ҽҺBұҵҲҴҽҶPҰFҶPҽӁҲҷҲҼҲҹҺPҲұҬPӒҷPҶBҵPҶBҴP ҫҺPKһҼҺBҷҲӀBҶҷPҭPҽҭҵBPһҷPҬFҷFPҭҺBҷҲӁFҷPҽҮҬP һҼҺҽӁBҬBҶPϖBҰFҶPҮBPҫҲҶҶҷPҭPҽҭҵBҲҶBұBҭҺBҷҲӀҽ PҫҲҶҴҺҽҰҷҲӀFҽҴPKҽKFҽҹҲһBҷ PҫұҲҺPҶҮBKFҬҲһҲҷB ҫPӁҷҲҿһҼҺBҷBҽҹҲһBҷҲҿҹҺҲұBҶBһҼBҵҷBҲKFҮҷBҴBҬҲ һҲҷҲҬBӒҴB ҭҺBҷҲӁҷBҬҺFҮҷPһҼҹPҬҺӂҲҷFPҶPҼBӁBҼҲҿ ҹҺҲұBҶBKFҮҷBҴBKFҹҺPҲұҬPҮҽPҫҲҶBPһҷPҬFҬBӒҴBҲӓF ҭPҬFҬҲһҲҷF5ҽҭҺBҷҲӁҷҽҬҺFҮҷPһҼҲҽұҲҶBҶPұBҹPҬҺ ӂҲҷҽPҶPҼBӁBҬBӒҴB

FҴBKF n ҫҺPKһҼҺBҷҲӀBҽҹҲһBҷPҭҶҷPҭPҽҭҵB pn PҫҲҶҶҷPҭPҽҭҵBҲ Mn ҹPҬҺӂҲҷBPҶPҼBӁBҹҺҲұҶF5BҮB

KF Mn = pn · HϖBҮ n ҷFPҭҺBҷҲӁFҷPҺBһҼF pn ҼFҰҲҴB

2πr Mn ҼFҰҲҹPҬҺӂҲҷҲPҶPҼBӁBҬBӒҴB M ҹBҶPҰFҶP

ҷBҹҲһBҼҲҮBKF M =2rπH.

BPһҷPҬҽҹҺFҼҿPҮҷPҭҲұҵBҭBӓBҶPҰFҶPҾҸҺҶҽҵҲһBҼҲҼҬҺӋFӓF

ϞүҸҺүҶҪ ϛPҬҺӂҲҷBPҶPҼBӁBҹҺBҬPҭҬBӒҴBKFҮҷBҴBKFҹҺPҲұҬPҮҽ PڈҲҶB PһҷPҬFҲҬҲһҲҷF

"ҴPһB B PұҷBӁҲҶPҹPҬҺӂҲҷҽPһҷPҬFҬBӒҴB BһB P ӓFҭPҬҽҹPҬҺӂҲҷҽ ҼBҮBKF

P =2B + M =2r 2 π +2rπH, ҼK

P =2rπ(r + H )

"ҴPһFҺBұҬҲKFPҶPҼBӁҹҺBҬPҭҬBӒҴBҮPҫҲ KBһFҹҺBҬPҽҭBPҷҲҴϐҽҰҲҷBKFҮҷFһҼҺBҷҲӀFҼPҭ ҹҺBҬPҽҭBPҷҲҴBKFҮҷBҴBKFPҫҲҶҽҴҺҽҰҷҲӀFPһҷP ҬF BҮҽҰҲҷBҮҺҽҭFKFҮҷBҴBKFҬҲһҲҷҲҬBӒҴBϐB ҴҵF ҶҺFҰBӀҲҵҲҷҮҺBһBһҼPKҲһFҲұҹҺBҬPҽҭBPҷҲ ҴBҲҮҬBҴҺҽҭB һҵ

PҬҺӂҲҷBPһҷPҬFKF

PҬҺӂҲҷBӀFҵPҭҬBӒҴBKF

"ҴPKFPһҷPҬBҹҲҺBҶҲҮFҶҷPҭPҽҭBPҽҹҲһBҷҽPһҷPҬҽҴҽҹF BҬҺҿҹҲҺBҶҲҮF KFҲһҼPҬҺFҶFҷPҲҬҺҿҴҽҹF ҴBҰFҶPҮBKFҹҲҺBҶҲҮBҽҹҲһBҷBҽҴҽҹҽ һҵ

һҼҺB ҷҲӀBҽҹҲһBҷPҭҹҺBҬҲҵҷPҭҶҷPҭPҽҭҵBҷFPҭҺBҷҲӁFҷP

FҴBKF r ҹPҵҽҹҺFӁҷҲҴPһҷPҬFҴҽҹF B M ҹPҬҺӂҲҷBӓFҷPҭPҶPҼBӁB B

PһҷPҬҽҹҺFҼҿPҮҷPҭҲұҵBҭBӓBҶPҰүҶҸҾPҺҶҽҵҲһBҼҲһҵFҮFӔFҼҬҺӋFӓF ϞүҸҺүҶҪ PҬҺӂҲҷBPҶPҼBӁBҹҺBҬFҴ

ϐBҴҵF M = 1 2 2rπs = rπs

ϐҸҴҪұ 0ҮҽұҶҲҶPPҮҹPҬҺӂҲҷFPҶPҼBӁBҹҺBҬFҴҽҹFҹPҬҺӂҲҷҽ PҶPҼBӁBӓFҷFҮP ҹҽҷF һҵ ϜBұҵҲҴBKFKFҮҷBҴBҹPҬҺӂҲҷҲPҶPҼBӁBҹҺBҬFұBҺҽڈӒFҷFҴҽҹF 0ڈүҵFҰҲҶPһB R Ҳ r ҺFҮPҶҹPҵҽҹҺFӁҷҲҴFҮPӓFҲҭPҺӓFPһҷPҬFұBҺҽڈӒFҷFҴҽҹF

FҴBKF s = |AB| ҮҽҰҲҷBҲұҬPҮҷҲӀFұBҺҽڈӒFҷFҴҽҹF B x = |VB| ҮҽҰҲҷBӓFҷFҮPҹҽҷF 5BҮBKF ҷBPһҷPҬҽ ҼFPҺүҶF ҹPҬҺӂҲҷBPҶPҼBӁBұBҺҽڈӒFҷFҴҽҹF M = Rπ(s + x) rπx.

ϔұһҵҲӁҷPһҼҲҼҺPҽҭҵPҬB VAO Ҳ VBC һҵF

ҮҲҮBKF x s + x = r R , Ҽӑ x = rs R r

BҮBKF

M = Rπ s + rs R r rπ rs R r , ӂҼPҹPһҵFһҺFӋҲҬBӓBҮBKF

M = πs(R + r). ✷

ϛPҬҺӂҲҷBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFKFҮҷBҴBKF P = B + B1 + M ҭҮFһҽ B Ҳ B1 ҹPҬҺӂҲҷFPһҷPҬBϖBҴPKF B = R2 π B1 = r 2 π ҴPҷBӁҷPһFҮPҫҲKBҾPҺҶҽҵB P = π(R2 + r 2 +(R + r)s)

0ҮҺFҮҲҹPҬҺӂҲҷҽ B ҹҺBҬPҭҬBӒҴB ҫ ҹҺBҬFҴҽҹF

ҹPҵҽҹҺFӁҷҲҴBPһҷPҬFDNҲҬҲһҲҷFDN

ϖBҴPһFҶFӓBҹPҬҺӂҲҷBPҶPҼBӁBҬBӒҴBBҴPһF B ӓFҭPҬBҬҲһҲҷBҹPҬFӔBҮҬBҹҽҼB ҫ ҹPҵҽҹҺFӁҷҲҴӓFҭPҬFPһҷPҬFҹPҬFӔBҼҺҲҹҽҼB

ϓBҴPҵҲҴPһFҶPҺBҹPҬFӔBҼҲҬҲһҲҷBҮBҼPҭҹҺBҬPҭҬBӒҴBҮBҫҲҹPҬҺӂҲҷBPҶPҼBӁB ҼBҴPҮPҫҲKFҷPҭҬBӒҴBҫҲҵBKFҮҷBҴBҹPҬҺӂҲҷҲҮBҼPҭҬBӒҴB

PҬҺӂҲҷBPһҷPҬFҹҺBҬPҭҬBӒҴBKF B BҹPҬҺӂҲҷBӓFҭPҬPҭPһҷPҭҹҺFһFҴB Qϔұ ҺBұҲҹPҬҺӂҲҷҽҬBӒҴB P ҴBPҾҽҷҴӀҲKҽPҮ

PҵҽҹҺFӁҷҲҴPһҷPҬFҹҺBҬPҭҬBӒҴBKF r BҹPҬҺӂҲҷBPҶPҼBӁBKFҮҷBҴBKFұҫҲҺҽҹP ҬҺӂҲҷBPҫFPһҷPҬF0ҮҺFҮҲҬҲһҲҷҽҬBӒҴB

PҬҺӂҲҷBKFҮҷBҴPһҼҺBҷҲӁҷPҭҬBӒҴBKF P =24π DN 2 0ҮҺFҮҲҹPҵҽҹҺFӁҷҲҴPһҷP ҬF

ҼҺBҷҲӀFҹҺBҬPҽҭBPҷҲҴBҲҶBKҽҮҽҰҲҷF a Ҳ b0ҮҺFҮҲҹPҬҺӂҲҷҽPҶPҼBӁBҬBӒҴB ҴPKҲһFҮPҫҲKBPҫҺҼBӓFҶҹҺBҬPҽҭBPҷҲҴBPҴPһҼҺBҷҲӀF a

0ҮҺFҮҲPҮҷPһҬҲһҲҷFҬBӒҴBҲҹPҵҽҹҺFӁҷҲҴBӓFҭPҬFPһҷPҬFBҴPKFӓҲҿPҬұҫҲҺ KFҮҷBҴҹPҵҽҹҺFӁҷҲҴҽҴҺҽҭBӁҲKBKFҹPҬҺӂҲҷBKFҮҷBҴBҹPҬҺӂҲҷҲҬBӒҴB

0ҮҵҲҶBҼҺFҫBҮBһFҷBҹҺBҬҲӀFҬҲҮҽҰҲҷFNҲҹҺFӁҷҲҴBDN B ϖPҵҲҴPKFҴҬBҮҺBҼҷҲҿҶFҼBҺBҵҲҶBҹPҼҺFҫҷPBҴPҷBӂBҬPҬFҲPҼҹBҼҴFPҮҵBұҲ ҶBҼFҺҲKBҵB

ҫ ϖPҵҲҴPKFҴҲҵPҭҺBҶBҫPKFҹPҼҺFҫҷPұBҫPKFӓFӀFҬҲBҴPKFұBҫPKFӓFҹPҬҺӂҲҷF PҮN2 ҹPҼҺFҫҷPҭҺBҶBҫPKF

ҬҲһҲҷBҶB H һҵ

PһҶBҼҺBKҶPҹҺFһFҴFPڈBҼFҵBһBҹҺPҶFҷӒҲҬPҶҺBҬҷҲ

PڈұҲҺPҶҷBPһPڈҲҷFҹBҺBҵFҵҷҲҿҹҺFһFҴBҹҲҺBҶҲҮFҲҴҽҹF ҹҺҲҶFҺ һҼҺ ҷBPһҷPҬ

ϐPҴBұ ϐPҹҽҷҲҶPұBҺҽҫӒFҷҽҴҽҹҽҮPҴҽҹFһBҬҺ

ҿPҶ V һҵ 5BҮBKFұBҹҺүҶҲҷBұBҺҽҫӒFҷFҴҽҹFKFҮ ҷBҴBҺBұҵҲӀҲұBҹҺүҶҲҷBҮҬFҴҽҹFoKFҮҷFһBҹPҵҽҹҺFӁ ҷҲҴPҶPһҷPҬF R ҲҬҲһҲҷPҶ |OV | = H + x ҲҮҺҽҭFһB

ҹPҵҽҹҺFӁҷҲҴPҶPһҷPҬF r ҲҬҲһҲҷPҶ |O1 V | = xϐBҴҵF

ҼҺBҰFҷBұBҹҺүҶҲҷBKF V = 1 3 R2 π(H + x) 1 3 r 2 πx = 1 3 π(R2 H + R2 x r 2 x) x H = r R r

3 πH R2 + x H (R2 r 2 ) = 1 3 πH R2 + r R r (R2 r 2 ) = 1 3 πH R2 + Rr + r 2 . ✷

ҹPҵҽҹҺFӁҷҲҴBPһҷPҬFDNҲҬҲһҲҷFDN ϖBҴPһFҶFӓBұBҹҺүҶҲҷBҬBӒҴBBҴPһF B ӓFҭPҬBҬҲһҲҷBҹPҬFӔBҮҬBҹҽҼB

ҫ ҹPҵҽҹҺFӁҷҲҴӓFҭPҬFPһҷPҬFҹPҬFӔBҼҺҲҹҽҼB

ϓBҹҺүҶҲҷBҬBӒҴB Ҵ

ϚҮҺүҮҲPҮҷPһұBҹҺүҶҲҷBҮҬBҹҺBҬBҬBӒҴBӁҲKFһҽҹPҬҺӂҲҷFPҶPҼBӁBKFҮҷBҴF ϖBҴPһFPҮҷPһFҹPҬҺӂҲҷFPҶPҼBӁBҮҬBҹҺBҬBҬBӒҴBӁҲKFһҽұBҹҺүҶҲҷFKFҮҷBҴF ϖBҴPһFҶFӓBұBҹҺүҶҲҷBҴҽҹFBҴPһF B ӓFҷBҬҲһҲҷBҹPҬFӔB n ҹҽҼB BҹPҵҽҹҺFӁҷҲҴPһҷPҬFPһҼBҷFҷFҹҺPҶFӓFҷ ҫ ҹPҵҽҹҺFӁҷҲҴPһҷPҬFҹPҬFӔB n ҹҽҼB BҬҲһҲҷBPһҼBҷFҷFҹҺPҶFӓFҷB 0ҮҺFҮҲұBҹҺүҶҲҷҽҹҺBҬFҴҽҹFҹPҵҽҹҺFӁҷҲҴBPһҷPҬF r =12 DNҲ H =18 DN ϓBҹҺүҶҲҷBҹҺBҬFҴҽҹFKF V =145 EN3 BҲұҬPҮҷҲӀBKFҹҽҼBҬFӔBPҮҹPҵҽҹҺFӁ ҷҲҴBPһҷPҬFϔұҺBӁҽҷBKҹPҬҺӂҲҷҽҴҽҹF

ϔұҬPҮҷҲӀBҹҺBҬFҴҽҹFҲҶBҮҽҰҲҷҽ l ҲұBҴҵBҹBһBҺBҬҷҲPһҷPҬFҽҭBPPҮ 30◦ 0ҮҺF

ҮҲұBҹҺүҶҲҷҽҴҽҹF

ώҲһҲҷBҲҮҽҰҲҷBҲұҬPҮҷҲӀFҹҺBҬFҴҽҹFPҮҷPһFһFҴBPBұBҹҺүҶҲҷBҴҽҹFKF 96π DN3 0ҮҺFҮҲҹPҬҺӂҲҷҽPҶPҼBӁB

ϐҬFҺBҬҷҲҹBҺBҵFҵҷFPһҷPҬҲҮFҵFҴҽҹҽҷBҼҺҲҮFҵBKFҮҷBҴҲҿұBҹҺүҶҲҷBϔұҺBұҲ ҺBһҼPKBӓBҺBҬҷҲҹҺFһFҴBPҮҬҺҿBҴҽҹFҹPҶPӔҽҬҲһҲҷFҴҽҹF PҵҽҹҺFӁҷҲҴPһҷPҬF ҬҲһҲҷBҲҮҽҰҲҷBҲұҬPҮҷҲӀFҹҺBҬFҴҽҹFһҽҼҺҲҽұBһҼPҹҷB ҹҺҲҺPҮҷBҫҺPKB PҬҺӂҲҷBҴҽҹFKF

0ҫҲҶPһҷPҬFҹҺBҬFҴҽҹFKF p BҮҽҰҲҷBҲұҬPҮҷҲӀF

0ҮҺFҮҲұBҹҺүҶҲҷҽҹҺBҬFұBҺҽҫӒFҷFҴҽҹFBҴPһҽҹPҵҽҹҺFӁҷҲӀҲӓFҷҲҿPһҷPҬB R =7 DNҲ r =2 DN BҹPҬҺӂҲҷB P =170π DN 2

PҬҺӂҲҷBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFKF P =308π DN 2 ҹPҵҽҹҺFӁҷҲҴPһҷPҬF R = 10 DNBҮҽҰҲҷBҲұҬPҮҷҲӀF l =17 DNϔұҺBӁҽҷBKұBҹҺүҶҲҷҽҴҽҹF

ϐҽҰҲҷBҲұҬPҮҷҲӀFҹҺBҬFұBҺҽҫӒFҷFҴҽҹFKF l =5 DN ҺBұҵҲҴBҹPҵҽҹҺFӁҷҲҴB PһҷPҬB R r =3 DNBҹPҬҺӂҲҷBPҶPҼBӁBKFҮҷBҴBKFұҫҲҺҽҹPҬҺӂҲҷBPҫFPһҷPҬF 0ҮҺFҮҲұBҹҺүҶҲҷҽҴҽҹF

ϔұҬPҮҷҲӀBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFұBҴҵBҹBһBPһҷPҬPҶҽҭBP α =60◦ PҵҽҹҺFӁ ҷҲӀҲPһҷPҬBһҽ R =9 DNҲ r =3 DN0ҮҺFҮҲұBҹҺүҶҲҷҽҴҽҹF

PҬҺӂҲҷBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFKF P =71π DN 2 ҲұҬPҮҷҲӀBҮҽҰҲҷF l =6 DN ұBҴҵBҹBһBPһҷPҬPҶҽҭBP α =60◦ 0ҮҺFҮҲұBҹҺүҶҲҷҽҴҽҹF

ϔұҺBӁҽҷBKұBҹҺүҶҲҷҽҹҺBҬFҴҽҹFӁҲKBKFҹPҬҺӂҲҷB 90π DN 2 BҲұҬPҮҷҲӀBKFұBDN ҮҽҰBPҮҹҺFӁҷҲҴBPһҷPҬF

ϖBҴPһFҶFӓBұBҹҺүҶҲҷBұBҺҽҫӒFҷFҴҽҹFBҴPһFӓFҷҲҹPҵҽҹҺFӁҷҲӀҲPһҷPҬBҹP ҬFӔBKҽҮҬBҹҽҼB BҬҲһҲҷBһҶBӓҲӁFҼҲҺҲҹҽҼB

PҵҽҹҺFӁҷҲӀҲPһҷPҬBҹҺBҬFұBҺҽҫӒFҷFҴҽҹFһҽ R Ҳ rϔұҬPҮҷҲӀBұBҴҵBҹBһBҺBҬ ҷҲPһҷPҬFҽҭBPPҮ 45◦ 0ҮҺFҮҲұBҹҺүҶҲҷҽҴҽҹF

ϖPҵҲҴPҵҲҼBҺBҬPҮFһҼBҷFҽҴPҾҽPҫҵҲҴBұBҺҽҫӒFҷFҴҽҹFBҴPKFҹҺFӁҷҲҴҮҷBDN ҹҺFӁҷҲҴPҼҬPҺBDN BҮҽҫҲҷBҴPҾFDN ҺBҬBҴҽҹBҹPҵҽҹҺFӁҷҲҴBPһҷPҬF R =10

DNҲҬҲһҲҷF H =24

DNҹҺFһFӁFҷB KFKFҮҷPҶҺBҬҷҲҹBҺBҵFҵҷPPһҷPҬҲ ҼBҴPҮBKFҼҲҶҹҺFһFҴPҶҹPҬҺӂҲҷBPҶPҼBӁB ҹҺFҹPҵPҬӒFҷB0ҮҺFҮҲұBҹҺүҶҲҷҽҮPҫҲKFҷFұBҺҽҫӒFҷFҴҽҹF ҺBҬBұBҺҽҫӒFҷBҴҽҹBһBҹPҵҽҹҺFӁҷҲӀҲҶBPһҷPҬB R =9 DN r =3

DNҲҬҲһҲҷPҶ H =8 DNҹҺFһFӁFҷBKFҹBҺBҵFҵҷPPһҷPҬҲҼBҴPҮBKFҹPҬҺӂҲҷBPҶPҼBӁBҹҺFҹPҵP ҬӒFҷB0ҮҺFҮҲұBҹҺүҶҲҷFҮҬBҮPҫҲKFҷBҮFҵB

Ҟ&Ϝ" ϔ 0 Ϟ"

ϝҾFҺBKFPҫҺҼҷBҹPҬҺӂҷBһҼBҵBPҫҺҼBӓFҶҹPҵҽҴҺҽҰҷҲӀFPҴPҹҺBҬFҴP KBһBҮҺҰҲӓFҷҹҺFӁҷҲҴҝҼҬҺҮҲҵҲһҶP ҼBҴPӋF ҮBKFһҾFҺBһҴҽҹһҬҲҿҼBӁBҴB ҹҺPһҼPҺBҴPKFһҽҷBҲһҼPҶҺBһҼPKBӓҽ

ϐүҾҲҷҲӀҲӑҪ

5FҵPPҭҺBҷҲӁFҷPһҾFҺPҶҷBұҲҬBһFҵPҹҼB

ϐҺҽҭҲҶҺFӁҲҶB ҵPҹҼBҹPҵҽҹҺFӁҷҲҴB r KFһҴҽҹһҬҲҿҼBӁBҴB X ҽҹҺPһҼPҺҽ ҼBҴҬҲҿҮBKFұBҽҼҬҺӋFҷҽҼBӁҴҽ O |OX | r5BӁҴB O KFӀFҷҼBҺҵPҹҼFBҺBһҼPKBӓF r KFҹPҵҽҹҺFӁҷҲҴҵPҹҼF

ҺҲһFӁFӓҽһҾFҺFҷFҴPҶҺBҬҷҲӁҲKFKFҺBһҼPKBӓFPҮӀFҷҼҺBҶBӓFPҮҹP ҵҽҹҺFӁҷҲҴB ҽҹҺFһFҴҽһFҮPҫҲKBҴҺҽҰҷҲӀB"ҴPҹҺFһFӁҷBҺBҬBҷҹҺPҵBұҲҴҺPұ

ϝFӁFӓFҶҵPҹҼFһBҺBҬҷҲӁҲKFKFҺBһҼPKBӓFPҮӀFҷҼҺBҶBӓFPҮҹPҵҽҹҺFӁ ҷҲҴBҮPҫҲKBһFҴҺҽҭ"ҴPҺBҬBҷҹҺFһFҴBҹҺPҵBұҲҴҺPұӀFҷҼBҺҵPҹҼF ҹҺFһFҴKF ҬFҵҲҴҲҴҺҽҭ ӁҲKҲKFҹPҵҽҹҺFӁҷҲҴKFҮҷBҴҹPҵҽҹҺFӁҷҲҴҽҵPҹҼF

ϐFPһҾFҺFҴPKҲPҮӓFPҮһFӀBҷFҴBҺBҬBҷ ҴPKBһҾFҺҽһFӁFҹPҴҺҽҰҷҲӀҲ ҷBұҲҬBһFһҾFҺҷҲһҬPҮ ҴBҵPҼB һҵ

ϖҺҽҭҴPKҲPҭҺBҷҲӁBҬBһҾFҺҷҲһҬPҮKF PһҷPҬB һҬPҮB FҴBҷPҺҶҪҵBҲұӀFҷҼҺB

ϐFPҵPҹҼFPҭҺBҷҲӁFҷһҾFҺҷҲҶһҬPҮPҶҲӓFҭPҬPҶPһҷPҬPҶҷBұҲҬBһFҵPҹ

ϐүҾҲҷҲӀҲӑҪ

ϐFPһҾFҺFPҭҺBҷҲӁFҷһBҮҬFҹBҺBҵFҵҷFҺBҬҷҲҴPKFһFҴҽһҾFҺҽҷBұҲҬBһF һҾFҺҷҲҹPKBһ һҵ

ϖҺҽҭPҬҲҴPKҲPҭҺBҷҲӁBҬBKҽҴҺҽҰҷҲҹPKBһһҽӓFҭPҬF

ϐFPҵPҹҼFPҭҺBҷҲӁFҷһҾFҺҷҲҶҹPKBһPҶҲӓFҭPҬҲҶPһҷPҬBҶBҷBұҲҬBһFҵPҹ ҼҲҷһҵPK һҵ

ҝҼFҿҷҲӀҲһFҵPҹҼBҴPҺҲһҼҲҴBPFҵFҶFҷҼҹҺҲұҭҵPҫҷPҶһҹBKBӓҽҮFҵPҬBҺB ұҷҲҿҶFҿBҷҲұBҶBҲҴPҮҴҽҭҵҲӁҷҲҿҵFҰBKBҭҮFұҷBҼҷPҮPҹҺҲҷPһҲһҶBӓҲҬBӓҽ PҼҹPҺBҼҺFӓBϐFҵPҬҲҵPҹҼFKBҬӒBKҽһFҲҽPҹҼҲӁҴPKҲҷҮҽһҼҺҲKҲ һPӁҲҬBҲҼҮ PұҷBҼPKFҮBҶFӋҽһҬҲҶҭFPҶFҼҺҲKһҴҲҶҼFҵҲҶBҮBҼFұBҹҺүҶҲҷFҷBKҶBӓҽҹPҬҺ ӂҲҷҽҲҶBҵPҹҼB

ҼҺPҭҲҮPҴBұҲҼFPҺүҶBPҹPҬҺӂҲҷҲһҾFҺFҲӓFҷҲҿҮFҵPҬB ҼFұBҹҺүҶҲ ҷҲҵPҹҼFҲӓFҷҲҿҮFҵPҬBұBҿҼFҬBKҽҬҲӂҲҷҲҬPҶBҼүҶBҼҲӁҴPҭұҷBӓBϓBҼPӔүҶҸ ҾPҺҶҽҵFҴPKFҮBӒFҷBҬPҮҲҶPҹҺҲҿҬBҼҲҼҲҫFұҮPҴBұB

ϐBҴҵF ҹPҬҺӂҲҷBһҾFҺFKFӁFҼҲҺҲҹҽҼBҬFӔBPҮҹPҬҺӂҲҷFҴҺҽҭBҲһҼPҭҹP

ҵҽҹҺFӁҷҲҴB

PҬҺӂҲҷBһҾFҺҷPҭһҬPҮB

P =2rπH,

ҭҮFKF H ҬҲһҲҷBһҬPҮB B r ҹPҵҽҹҺFӁҷҲҴһҾFҺF

PҬҺӂҲҷBһҾFҺҷPҭҹPKBһB

P =2rπH,

ҭҮFKF H ҬҲһҲҷBҹPKBһB B r ҹPҵҽҹҺFӁҷҲҴһҾFҺF

ϓBҹҺүҶҲҷBҵPҹҼFҹPҵҽҹҺFӁҷҲҴB r V = 4 3 r 3 π

ϓBҹҺүҶҲҷBҵPҹҼҲҷPҭPҮһFӁҴB

ҭҮFKF H ҬҲһҲҷBPҮһFӁҴB B r ҹPҵҽҹҺFӁҷҲҴҵPҹҼF

PҮҹPҬҺӂҲҷPҶҵPҹҼFҹPҮҺBұҽҶFҬBҶPҹPҬҺӂҲҷҽһҾFҺFҴPKBKFPҭҺBҷҲӁB ҬB

PҬҺӂҲҷBҮFҵBҵPҹҼFKFҹPҬҺӂҲҷBҹPҬҺӂҲҴPKBKFPҭҺBҷҲӁBҬB BҹҺҲҶFҺ

ҹPҬҺӂҲҷBҵPҹҼҲҷPҭPҮһFӁҴBKFұҫҲҺҹPҬҺӂҲҷBPһҷPҬFҲһҾFҺҷPҭһҬPҮBҴPKҲKF PҭҺBҷҲӁBҬBKҽ

14 DN

ϝҾFҺB ҵPҹҼB KF ҽҹҲһBҷB ҽҹPҵҲFҮBҺBҴPҮPҮҲҺҽKFһҬFһҼҺBҷFҼPҭҹPҵҲF

KFPҴPҴҽҹFBҴPһBҮҺҰҲҴҺҽҰҷҲӀҽPһҷPҬFҲӓFҷҬҺҿϓBҬBӒBҴ PҮҷPһҷPҴҽҹҽ ҽ

ϝҾFҺB ҵPҹҼB KFҽҹҲһBҷBҽҬBӒBҴBҴPҮPҮҲҺҽKFPҫFӓFҭPҬFPһҷPҬFҲPҶP

ҺҲҼPҶFҮPҮҲҺҽKFPҶPҼBӁҹPKFҮҷPKҴҺҽҰҷҲӀҲϝҾFҺB ҵPҹҼB KFҽҹҲһBҷB ҽҴҽҹҽBҴPҮPҮҲҺҽKFӓFҷҽPһҷPҬҽҲPҶPҼBӁ ҹҺҲӁүҶҽҮPҮҲҺҽKFPҶPҼBӁҹPҴҺҽ ҰҷҲӀҲ

ҽҭҵB OO1 B һҵFҮҲҮBKF

P = πrl + πr 2 = πR TJO 2α R TJO

0ҴPҴPӀҴFKFPҹҲһBҷBһҾFҺB0ҮҺFҮҲұBҹҺүҶҲҷҽPҮһFӁҴBҴPKҲPҮҹҺҲҹBҮBKҽӔF ҵPҹҼFPҮһFӀBҺBҬBҷҴPKBһBҮҺҰҲKFҮҷҽһҼҺBҷҽҴPӀҴFϐҽҰҲҷBҲҬҲӀFҴPӀҴFKF a

ҾFҺBҴPKBPҭҺBҷҲӁBҬBҵPҹҼҽұBҹҺүҶҲҷF V =288π ҹҺFһFӁFҷBKFKFҮҷPҶҺBҬҷҲ ҷBҮҬBһҬPҮBӁҲKFһFҹPҬҺӂҲҷFPҮҷPһFҴBPϖBҴPһFPҮҷPһFұBҹҺүҶҲҷFPҮҭPҬB ҺBKҽӔҲҿҵPҹҼҲҷҲҿPҮһFӁBҴB

PҬҺӂҲҷBҵPҹҼҲҷPҭPҮһFӁҴBҲұҷPһҲ 144π BҹPҵҽҹҺFӁҷҲҴҵPҹҼFKF R

0ҮҺF ҮҲҬҲһҲҷҽPҮһFӁҴB

PҬҺӂҲҷBҵPҹҼFҹPҵҽҹҺFӁҷҲҴB r ҹPҮFӒFҷBKFҷBҹFҼKFҮҷBҴҲҿҮFҵPҬBҹBҺBҵFҵ ҷҲҶҺBҬҷҲҶBҝҴPҶPҮҷPһҽKFҹPҮFӒFҷҹҺFӁҷҲҴҵPҹҼFҷPҺҶҪҵBҷҷBҼFҺBҬҷҲ

PҬҺӂҲҷBһҾFҺҷPҭҹPKBһBKFҮҷBҴBKFҹPҬҺӂҲҷҲPҷPҭҮFҵBPҶPҼBӁBҬBӒҴBPҹҲһB ҷPҭPҴPҵPҹҼFҴPKҲһFҷBҵBұҲҲұҶFӋҽҺBҬҷҲPһҷPҬBPҮҭPҬBҺBKҽӔFҭҵPҹҼҲҷPҭһҵPKB

ϔұҺҪұҲҽҺҪҮҲӑҪҷҲҶҪҽҭҵҸҬү

ϔұҺҪұҲҽһҼүҹүҷҲҶҪҽҭҵҸҬү

ϔұҺҪұҲҽһҼүҹүҷҲҶҪҽҭҪҸҴҸӑҲӑүҷҪҹҸҺүҮҪҷҽҭҵҽ

ϔұҺҪұҲҽҺҪҮҲӑҪҷҲҶҪ B ҽҭҵҸҬүӑүҮҷҪҴҸҴҺҪҴҸҹҺҪҬҸҽҭҵҸҭҼҺҸҽҭҵҪ

ҰҷҲӀҪҹҸҵҽҹҺүӁҷҲҴҪӁҲӑҲӑүӀүҷҼҪҺҽ

ҴҸҸҺҮҲҷҪҼҷҸҶҹҸӁүҼҴҽϞҪӁҴҪ

ҼҺҲӑһҴҸӑҴҺҽҰҷҲӀҲ ұҸҬүһү

ҴҪ һҵ ϙҪҼҺҲҭҸҷҸҶүҼҺҲӑһҴҸӑҴҺҽҰҷҲ

# « OA, # « OM )

ҽ o***ҴҬҪҮҺҪҷҼҽ ҪҴҸҹҸҵҽҹҺҪҬҪ OM ҹҺҲҹҪҮҪ***ҴҬҪҮҺҪҷҼҽ o*7ҴҬҪҮҺҪҷҼҽ ҪҴҸҹҸҵҽҹҺҪҬҪ OM ҹҺҲҹҪҮҪ*7ҴҬҪҮҺҪҷҼҽ όҴҸһүҹҸҵҽҹҺҪҬҪ

ҭ 450◦ =270◦ +( 2) · 360◦ ; δ

ϓόϐόҠϔ

ҺүҮһҼҪҬҲҷҪҼҺҲҭҸҷҸҶүҼҺҲӑһҴҸӑҴҺҽҰҷҲӀҲҸҺҲӑүҷҼҲһҪҷҽҭҪҸ

ҽ ҼҪӁҴҲ M ҽ

ҷҪ y Ҹһҽ һҵ ϞҪӁҴҪ M ҹҺҲҹҪҮҪ*ҴҬҪ

ҽ ҲӓүҷүҴҸҸҺҮҲҷҪҼүһ

ҸڈүҵүҰҲҶҸ ҽҭҪҸ ( ( # « OM, # « OA ) ҼҪҮҪӑү

( # « OA

үҴҪһҽ (x0 ,y0 ) (x0 < 0,y0 < 0) ҴҸҸҺҮҲҷҪҼүҼҪӁҴү

O һҵ ϞҪӁҴҪ M

ҲҲҶҪҴҸҸҺҮҲ ҷҪҼү ( x0 , y0 )όҴҸһҪ α ҸұҷҪӁҲҶҸ ҽ

( # « OA # « OM ) ҼҪҮҪӑүұڈҸҭһҲҶү

( # « OA # « OM )= α ҼҸҭҪӑү

DPT β = x0 = ( x0 )= DPT α; TJO β = y0 = ( y0 )= TJO α;

UH β = TJO β DPT β = TJO α DPT α = UH α; DUH β = 1 UH β = 1 UH α = DUH α.

ұҪһҬҪҴҲ ҽҭҪҸ α 0 <α< π 2

DPT(π + α)= DPT α

TJO(π + α)= TJO α

UH(π + α)= UH α

DUH(π + α)= DUH α

(180◦ +22◦ )= DPT 22◦ ≈−0, 92718; TJO 5

(180◦ +63◦ )= TJO 63◦ ≈−0, 89101; UH 4π 3 = UH π + π 3 = UH π 3 = √3; UH 200◦ = UH(180◦ +20◦ )= UH 20◦ ≈ 0, 36397; DUH 235◦ = DUH(180◦ +55◦ )= DUH 55◦ ≈ 0, 70021; DUH 7π 6 = DUH π + π 6

) ҽ

ҽ (x0 ,y0 ) (x0 > 0,y

ҽ ҲҲҶҪҴҸҸҺҮҲҷҪҼү (x0 , y0 )όҴҸһҪ α ҸڈүҵүҰҲҶҸ ҽҭҪҸ ( # « OM, # « OA) ҼҪҮҪӑүұڈҸҭһҲҶүҼҺҲӑү ( # « OA # « OM )= αϔұҼҸҭҪһҵүҮҲ

DPT β = x0 = DPT α; TJO β = y0 = ( y0 )= TJO α; UH β = TJO β DPT β = TJO α DPT α = UH α; DUH β = 1 UH β = 1 UH α = DUH α;

DPT(2π α)= DPT α

TJO(2π α)= TJO α

UH(2π α)= UH α

DUH(2π α)= DUH α

Ҫ TJO 5π 6 ڈ TJO 1Ҭ DPT 3π 4 ҭ DPT 0, 2Ү TJO( 2)Ӌ DPT( 3)

Ҫ TJO 178◦ ڈ DPT 359◦ Ҭ TJO 210◦ ҭ DPT 300◦ Ү TJO 288◦ Ӌ DPT( 91◦ )

ϚҮҺүҮҲұҷҪҴ Ҫ UH 175◦ ڈ UH 269◦ Ҭ UH( 95◦

ҭ UH 1Ү UH 0, 2◦ Ӌ UH( 2◦ )

ϚҮҺүҮҲұҷҪҴ Ҫ DUH 359◦ ڈ DUH 200◦ Ҭ DUH( 182◦

ҭ DUH 1 Ү DUH 1, 5◦ Ӌ DUH( 2◦ )

Ҫ TJO α> 0 Ҳ DPT

DPT(x0 + T ) = DPT

ϓόϐόҠϔ

ҽҷҪӑ B TJO 390◦ ڈ DPT 420◦ Ҭ TJO 540◦ ҭ DPT 7π 3 Ү TJO 15π 4 ү DPT 19π 6

ϔұҺҪӁҽҷҪӑ Ҫ DPT( 720◦ )ڈ DPT( 780◦ )Ҭ TJO( 405◦ ) ҭ TJO 9π 4 Ү DPT 13π 3 ү TJO 17π 6

ϔұҺҪӁҽҷҪӑ Ҫ UH 390◦ ڈ UH 540◦ Ҭ UH( 810◦ ) ҭ UH 7π 3 Ү UH 17π 4 Ӌ UH 35

ϔұҺҪӁҽҷҪӑ

Ҫ DUH 450◦ ڈ DUH 750◦ Ҭ DUH( 1110◦ )

ҭ DUH 11π 3 Ү DUH 23π 4 Ӌ DUH 33π

һүҶҸҼҪӁҴү C0 ,C1 ,C2 ,...,C11 ӁҲӑү

(

ҷҽҵүҾҽҷҴӀҲӑүһҽ x = kπ (k ∈ Z) ҷүҶҪҷҲҶҪҴһҲҶҪҵҷҲҿҷҲҶҲҷҲҶҪҵҷҲҿҬҺүҮҷҸһҼҲ

, +∞)

DUH x һҼҪҵҷҸҸҹҪҮҪ

Ҫ y = UH x +1ڈ y = DUH x 1Ҭ y = UH xҭ y = DUH x +1

Ҫ y = TJO x +2ڈ y = TJO x 1Ҭ y = TJO( x)

Ҫ y = UH x 2ڈ y = UH x +1Ҭ y = UH( x)

Ҫ y = DUH x +2ڈ y

y = 2 TJO xڈ y = 1 2 TJO x

y = TJO 3xڈ y = TJO x 3 Ҭ y = TJO 2x ҭ y = TJO 3x 4 Ү y = TJO(2x 5)Ӌ y = 2 TJO(3x +4)

όҴҸӑү M (x0 ,y0 ) ҹҺҸҲұҬҸӒҷҪҼҪӁҴҪ

x 2 0 + y 2 0 =1. ϖҪҴҸӑү x0 = DPT α Ҳ y0 = TJO

DPT 2 α + TJO2 α =1 ұҪһҬҪҴҲ ҽҭҪҸ α

Ҫ (2+ TJO α)(2 TJO α)+(2+ DPT α)(2 DPT α)=7

ڈ DUH α + TJO α 1+ DPT α = 1 TJO α

Ҭ 1 2 TJO α DPT α TJO α DPT α = TJO α DPT α

ҭ 1 TJO2 α 1 DPT2 α = 1 UH2 α

Ү DPT2 α TJO2 α DUH2 α UH2 α = TJO2 α DPT 2 α

Ҫ 1+2 TJO x DPT x (TJO x + DPT x)2

ڈ TJO2 x DPT2 x +1 TJO2 x

Ҭ 1+ TJO x DPT x · 1 TJO x DPT x

ҭ 1 4 TJO2 x DPT2 x (TJO x + DPT x)2 +2 TJO x DPT x

Ү 3(DPT 4 x + TJO4 x) 2(DPT6 x + TJO6 x)

(α + β)= DPT α · DPT

όҮҲӀҲҸҷүҾҸҺҶ

ҺҲҶүҺ ҝҹҺҸһҼҲҶҸҲұҺҪұү

B DPT(60◦ α)+ DPT(60◦ + α)

ڈ TJO(30◦ + α)+ TJO(30◦ α)

Ҭ DPT(α + β)+ TJO α TJO β DPT(α β) TJO α TJO β

ҭ TJO(α + β) DPT α TJO β TJO(α β)+ DPT α TJO β Ϝүӂүӓү

B DPT(60◦

(30

= DPT α · DPT β DPT α DPT β =1(DPT α · DPT β =0);

TJO(α + β) DPT α TJO β TJO(α β)+ DPT α · TJO β = TJO α DPT β + DPT α TJO β DPT α TJO β TJO α · DPT β DPT α · TJO β + DPT α · TJO β = TJO α · DPT β TJO α · DPT β =1(TJO α · DPT β =0).

Ҫ UH x = √3ڈ UH x =

Ҭ UH 2x = 1ҭ UH 3x =0

Ү UH x π 6 =

ڈ DUH x =1

Ҭ DUH 3x =1ҭ DUH 5x =0

Ү DUH x π 2 =1Ӌ DUH 2x π

ڈ2 b TJO α a =1ϞҪҮҪӑүTJO β = b TJO α a

Павлов

Росанда Вучићевић

Милан јелановић

Александар Радовановић

Жељко Хрчек

Корице м ранислав Николић

Милева Радосављевић Обим: 17,5 штампарских табака Формат: 16,5×23,5 cm

Тираж: 1500 примерака

Rукопис предат у штампу 2014. године. Штампање завршено 2014. године.

Рукопис предат у штампу августа 2023. године. завршено септембра 2023. године.

Службени гласник, еоград

Математика за 2. разред средње школе - 22179

MATEMATIKA

za II razred sredwe {kole

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ϓ"ϐ"Ҡϔ

ϓ"ϐ"Ҡϔ

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