__________________________________________________________________275

 1 1.  8.  15. 

2.  9.  16. 

3.  10.  17. 

4.  11.  18. 

5.  12.  19. 

6.  13.  20. 

7.  14.  21. 

 2 1. 36π π  ..

2. 96  .. 3.

L 1+

5. 13 : 18 9. 2a2(π π-2)  

2 π

! 1     3 ..

A2

10 ..    B =

1+ 3

3VH 2 πR 2

6. 4   7. 910  .. 8. 249,900 ..2 10. 0.75   11. 27   

 1

A1 O. 144° O B

L

4.

D 144 98 2 2 × π × ( 10 − 3) = π o × πr = 360 5 360

276_____________________________________  !"#$ " 
"% 98 147 π= π 5 5 144 98 98 102    A2 = × π × 10 2 − π = 40π − π = π 360 5 5 5 147 102 249     = π+ π= π 5 5 5

   A1 =    -    B = 49π −




! 2    

   = πr 2 h =

!

22 7 7 × × × 10 = 385 7 2 2

1 ×   # × $ 3 1 = × 4 × 4 × 6 = 32 ! 3

 " =

385 = 12.03 32

%&

  % ' "(" $)" 12  ! 3     100-2x


A = x ( 100 - 2x ) = 100x - 2x2 % 3 ≤ x ≤ 10 %(" 300 ≤ 100x ≤ 1000

x

.. ( 1 )

. ( 2 )

 9 ≤ x 2 ≤ 100 −100 ≤ − x 2 ≤ −9 −200 ≤ −2 x 2 ≤ −18



. ( 3 )


(2)+(3) 100 ≤ 100x − 2 x ≤ 982 ! 4     *+$ + ,- - a ..  , b .. "  b = 2a .", 5 ..  5 ..   $ + ,/  75 .. ( b - 5 )( a + 5 ) - ab = 75   b = 2a %(" ( 2a - 5 )( a + 5 ) - a×2a = 75 2a2 + 10a - 5a - 25 - 2a2 = 75 5a = 100 "  a = 20 "  ,"$ + ,- -0 20  ,",$ + ,- -0 40  2

  $ + ,  =

1 × 20 × 40 = 400 2



%("





   ___________________________________________________________________277 ! 5     *+" '*.12$ h ..

(!   + ,2 =

1 × π × 10 2 × 40 2

= 2000 π !  '*.12 = 40 2 × h − π × 10 2 × h %(" 1600h - 100hπ = 2000π h

! 6     A

=

2000π 20π =  1600 − 100π 16 − π

*+! OA = r  ON %(" ON = r ^

M

N 60 o 30°

O    = =




%$+ , MON cos MON = " 

OM 1 = ON 2

^

MON = 60 o



^

NOB = 30 o

B    ∆MON +   $ + ,#3 ONB 1 r 1 π × × r sin 60 o + × r 2 × = 0.48r 2 2 2 2 6

= 0.48  2  $ + ,%)$  ," r +2,




! 7     ",2"*4&" %(" /0-2  "$ ,

278_____________________________________  !"#$ " 
"%

C

8

D 3

8

8 A 5

8

B 3 R

3 Q

5 a P

% $+ , RPQ ,$+ , BPA a a+3 = ⇒ a=5 5 8

%("

AP = 5+3 = 8  BP = 8 $"2$+ , ABP 0$+ ," 2 %%)" P $  AB %)" S *+ T 0%)""$ ,) AC  BD %(" PT 0$2$ "  ST  

P

8

D 8 A

4

T

S 4

C 8

B

PS = 8 2 − 4 2 = 4 3 , ST = 4 PT =

( 4 3)

%("

2

− 42 = 4 2 1  " PABCD = × 8 2 × 4 2 3

!+2,

+  "%&("* '" , $ + ,%)$," 5 +2, "  "0

 %#0


   ___________________________________________________________________279 1 2 5 ×5 × 2 3 2

!+2,

"  

 1  3

1 3

=  × 64 × 4 2  −  × 25 ×

5 2  129 2 !+2, = 2  2




! 8     .$1 %)"!,  $  %("$ + ,%)$"

A

M

H

E

B

F

N

O

D G

L

P

*+ O 0%)"!,/% 0%)""$ ,)$ + , %)$ "%$1 "2     =    ∆ MON −

1 1   ! a +2, -    4 4 , OH = 2a − a = ( 2 − 1) a

ON = AO 2 + AN 2 = a 2 + a 2 = 2a 1 1    ∆ MON = × AO × MN = a × 2a = a 2 2 2 2   ! a +2, = πa

  

= π( 2 − 1) a 2

"    

1 1 4 4  2 − ( 2 − 2 )π  2 a 2 2 − 1) π =  2  

=

(

a2 4−π−( 4

2

= a 2 − πa 2 − π( 2 − 1) a 2

)

2

+2,




280_____________________________________  !"#$ " 
"% ! 9   



C E

D O

A

B F

*+ O 0%)"!,  $ %)"$-$,2  D, E  F  OD, OE, OF 3", $$$-$ %(" CD = BD, AE = CE , AF = FB %(" CE = CD, BD = BF "  AB = BC = AC $"2 ABC 0$+ ," 2  OC %(" OC = R  % OD⊥BC  OD = r %(" CD = R 2 − r 2   BC = 2 R 2 − r 2

%("$2$ CF =

3 × 2 R2 − r2 = 3 × R2 − r2 2

   $+ , ABC   1 × 2 R 2 − r 2 × 3 × R 2 − r 2 = 3( R 2 − r 2 ) = 2

3( R 2 − r 2 )

+2,


! 10    

C

A

B

*+ A  B 0%)"!,  C 0%)""  $ %(" ABC 0 $+ ," 2

1 π 2 πr 2    ABC = × × r = +2, 2 3 6 1 3 3 2   $+ , ABC = × r × r = r +2, 2 2 4

%("  $2  = 2 (    ∆ ABC ) + 4 (    ABC -    ∆ ABC )
   ___________________________________________________________________281 = 4 (    ABC ) - 2 (    ∆ ABC )

 4π − 3 3  2  πr 2   3 2  r  =   − 2 r 6  6   4   

= 4

+2,




! 11     *15 2" AD  BC ( %)" E %("

E D

3/2

C

ED EA = 3 2 2 3 ED = ( 1+ ED) 4

ED = 3 "  AE = 4

2 A

B

  "%+)$ + , ABCD  AD  2 -2 3 +2, 2 2  1  1  3 2     π × 2 × 4 − × π× × 3    = 3  3 2  16π 9π 37π = − = !+2, 3 4 12

,! 2 +2, !




! 12    

A E B C

F

D

 % ∆ ABD  ∆ AED  %)"," A 2  #,2$" , $"2  $2$ 2     ∆ ABD :    ∆ AED = BD : ED = 2 : 3    ∆ ABD =

2    ∆ AED 3

* '" ,    ∆ BCD :    ∆ CDE = 2 : 3    ∆ 6BCD =

2    ∆ CDE "  3

282_____________________________________  !"#$ " 
"% (    ∆ 6BCD +    ∆ BCD ) : (    ∆ 6AED +    ∆ CDE ) = 2 : 3 $"2    ABCD :    ∆ ADC = 2 : 3

   ∆ ADC =

3 3    ABCD = × 36 = 54 +2, 2 2




! 13    

D

C E F 45° 60° 1-a P

A

45° a

B ^

 % ABE 0$+ ," 2 %(" ) EAB = 60 !  FP⊥AB

%("

^

^

PFB = PBF = 45o

*+ PB = a %(" PA = 1-a  PF = a

a 3 = 3 ⇒ a= 1− a 1+ 3 1 3 3− 3 %("   $+ , AFB = × ×1 = +2, 4 2 1+ 3

%&*$+ ,) APF

tan 60 o =

! 14     *+ r  !  $, =

a = 3a tan 30 o




%&$+ , ABC %("

3",!,$+ ,) ADO %("

A 30°

D• O B

|------a----|C

r 3 3a ⇒ r = a o +r = 3 sin 30 3 4  3  4 3πa 3 %("   = π a  = !+2, 3  3  27





   ___________________________________________________________________283 ! 15   



B A C *+ A, B  C 0%)"!,  $  AB, BC, CA $+ ," 2 ABC %("   $+ , ABC  2

%2*+"

1 × 2 r × 2 r sin 60 o = 3r 2 2

60 πr 2     $0 3 × πr × = 360 2 π 2  "   &&  3 − r +2,  2 2




! 16     $)*+3+* , $0 h  $-2!,0 d 

 %3+*$ $-2!,,

%("$-2!,3+*$0

2 $-2!,3+* , 3

2 d  3

*+ V1 03+*$  V2 03+* , %$   = πr 2 h 2

2

 2d  d   × 3h  V2 = π ×   × h 3× 2 2 2  2d   × 3h 1 π × V 3× 2 3 4 %(" 1 = = = 2 1 3 V2 d  π ×  × h 4 2

V1 = π × 

2 V1 = 32  = 32,000 ! " 

32,000 4 = V2 3

⇒ V2 =

32,000 × 3 = 24,000 ! 4

2, '$  5  284_____________________________________  !"#$ " 
"%

32,000 × 5 = 800  200 24,000 , '$3+* ,(" = 600  200

, '$*$("

" 2, '$ ("  $  800 + 600 = 1,400 




! 17      %  $-2!,0 d  $+  = 4 d  π  3 2

4 3 πr 3

"   '+"%/ 0

3

 *1/+/   ,*$2*,%"

$)$2$,0 h  1 ×   2 2 3 1 d  1 4 d π  h = × π  %(" h = d 3 2 2 3 2

, =




! 18    

D

C

A

B H

E

G F

*+2"$ + ,%)$, a +2,

  - 6 "0 = 6a2 = 216      ∆AEG =

1 × AE × EG 2

 % ∆EFG 0$+ ,)  EF = FG = x = 6

%("

∆AEG  AE 0$2$  EG 0#

EG = 6 2 + 6 2 = 6 2

"     ∆EFG =

1 × 6 × 6 2 = 18 2 +2, 2





   ___________________________________________________________________285

! 19   

D

D

E

B

F

 %

2

C

$),"$ + ,%)$ BDEF AB = a + 8  BC = a + 2

," a 

%("

1 2 1   $+ , ADE 0 ⋅ a ⋅ 8 2 1   $+ , FCE 0 ⋅ 2 ⋅ a   $ + , BDEF = a2 2

  $+ , ABC 0 ( (a + 8) ( a + 2)

2  $+ , ABC =   $+ , ADE +   $+ , FCE +   $ + , BDEF %(" 1 1 1 ( a + 8)( a + 2) = ⋅ a ⋅ 8 + ⋅ 2 ⋅ a + a 2 ⇒ a 2 + 10a + 16 = 8a + 2a + 2a 2 2 2 2 ⇒ a 2 = 16 "  a = 4 1 %("  $+ , ABC 0 × ( 4 + 8)( 4 + 2) = 36  2




! 20    

A P

B

E

S

D

Q

R

C

286_____________________________________  !"#$ " 
"%

%&#/ 0$+ ,+%  AD ⊥ BC %)" D %(" AD = 52 − 32 = 4  2 AD / / PS %(" ∆ABD ~ ∆PBS . PS , a   % 1 2 AD BD 2 = PS BS

a  2 4 3 %(" = ⇒ a a 3− 2

SD = SR %(" SD ,

a = 24 

$"2/ PQRS =24×2.4×20 =115.2 !




! 21     2

1 c   /  # AB = π  +2, 2 2 2 1 b     /  # AC = π +2, 2 2 2 1 a   /  # BC = π  +2, 2 2 1   $+ , ABC = ab +2, 2 2 2 2 1 a 1 b 1 c 1  π  + π  −  π  − ab    & = 2 2 2 2 2 2 2 

=

 a 2 + b2 − c2  1 π  + ab 8   2

2$+ , ABC 0$+ ,)/ ) C 0) 1 2

%(" a2 + b2 = c2      0 ab +2,








   

 

  

   !  "  # $!%  
& '(  & %)
 *
 
+ & 
,-