Сборник задач ТСПП

ρ

2

–( ρ

=

=

= /3,

$ .

k

2

k

–

1.2/.8. '$

1,/43, (

%

– 1,/3.0-%

( ) % ,(

( '$

2,/53, $ – #1,5?

,( (

1,/31?

1.2-.9(.

$ ,( (

'$ 1,/53, $

– 1,5?

1.2.9.$ ( ρ = ρ k= 1,5=1, /3.

1.2).!10.! ' 3 (

% '

$ '

15 60

50 ) % ! '

G =

200'0

10%.

:)( % ! ; )%

! ! ;) ! ;) ! .

1.2

.10.

G= 2000= 2 ;

–! '

m=

g

9,81

+

+

–( !

S=

a

b=h0,50 0,600,15=

2;

2

3;2

– $ !

= = 3 0,0825=

3;

V

–

0,204 /3;

( ! ρ

. =

m=

0,2475=

V

0

– ! ( ! :

ρ

− ρ

=

â.ë âò .

ñρâò = ρ

.

− =

=

W

ñ. âò

(1W ) 0,85

ρ â.ë âò

âë âò

–% ! !

q=

m=

204= /;

L 3

– ! G=

Vρ

= g0,2475 0,76=

ñ

æ

.ñ âò

=19 .

– ! m=

m

.m.= 204−

/3;

2 ;

1.2+.11. 1.2.10 ,(

,				1.2.3.
											1.2.3
& !
! ,	4		6	8	6	7	8		12	13		10
! ,2	1	2	3	2	7 7	2	2	2	9	10 10
!
,			50×3060×4040×3035×1542×2855×3065×2							0		×2040×60×15
,

11

		1.+3.
+ –
(	! ).

.

1.3.1. $ 0 1, 2, 3

! ( ' ! . 1,. 2 . 3.

!				– $		( ).
# !							30,	50!, .20
, % ! ,(
						100,	2070,0 $
				0,3;	0,5" ''; 0,7. $
3 ! .! ,(
( $					)%
! .
- ,( ! $
			1:1,5:2,5?
	1.3.1!. , ! $ % ! :
+ 1= ×30×5 18×				0,3× 100 = ;81000
+ 2= ×50×5 18×				0,5× 200 = ;450000
+ 3= ×20×5 18×				0,7× 70 = .88200
	:( % ! ,
% $ ! 1,					3-(. ! $

1:1,5:2,5!, $ , ! ! .

( ! ) '$ .

1.34.2. ! ( )'

$ ! , $ %

, % – ' , ! .-

( ) $ ,( ! ,(

% ! ?

,1.3.2. ! % % ! ,

( (

! .) % % ! %(

)-.( ! ,( !

' ,% ( ) , (

% ! .

12

2.

* ,

, : (. 3)

–

, ;

; ; .

: (

)

.

“

”! .

(

),"

:

t ;

(

) ;t

(

)

λ =

1

, / ;

t

−λ

;

t,

P(t)=!

,k

+

=

,

kkã ðåìk

"

(

. # , ,

!

.

2.1. ,

n

( ).

,

:

– t. t, ;

	– (			) $ , / ;
	– t, ;
	–				,
	! k					k.
			....2, .... )(				.
	" t1,t2, t1, t
	"		2.1			( )
.
	!2.1.		:
	–!
			n		n
			(		( t

	t	=	= 1 t	,			t	=	= 1	,
			i n						i n
n–	;					1
–				λ =			/ ;

t

– Pt= −λ();

*# .

13

–!

=		t		=	t
	(t
			;		(	t.
k		+		k	t+
2.$2.							(4.

) : $ 1= 1/8$ 2=,1/12$ 3= ,1/4$ 4=,1/6 ( )

4= 0,7.

t1 = 2t,2= 0,5t3 =, 1t,

, ,!

, 1, 2, 5 10 .% ,

? t ,

, 0,99, 0,75, 0,5.

#2.2. , ,

! , !

	+		+		+
λ = λ + λ		λ		λ =		1
ëö 1	2		3	4 8

1			1			1
	+		+		=	4	/ .
12			6

" !

=

=

=

.

.

λ 1

15241

t

P

e− 15

;

2

&

1

– )1(=

241=

1

=

,054

–

,062

P

e152

1

15

P e−15

.

−

1,24

−

3,1

2,6

(2)

24 =

; 5 –P(5)=

=

=

e245=

1=

; 10–(10)=

2410=

1=

,002

e

e

' !

k.

=

3

.

k1

2k

! 1=

t1

=

8

= 0

=

t2

=

12

= 0,

1

1

8 2

; 2

2

2

12+ ,05 ,

k t

+

t

+

+

t

k t

=

t3

=

4

=

0

=

t4

=

6

= 0

3

3

3

4 1

;

4

4

+

4

6+

0,7 .

+

t

+

t

"

k t

k t

k = 0,8

0,96=0,

.

−

3

( t P(t= )0,990,99=:e81t 1=t;

.

− 3

–

0,750,75=:e82t 2=t;

.

−

3

3=t; .

–

0,50,5=: e8t3

2.)3. ! !

!(

)

3-

."

10

30

.(

– 1.5

.*

–’ , +

8

, 22

.

,

t,t,$ ,k

k

% (t)

t,

8,

20,

40,

100 ., %

=f(t).

14

'2.3.

3

t= ×3 2×2 8

= 52.8

(

3

t. = 15× 3 .×

22 = ×0,25×3 22=16,. 5

-

t= −t . = 528 – 16,.5 = 511,5

-

t

=

3

= 3,

10,

1

– 3.0

-

−

−

t

=

(

=)(511,5=

30)= 48,1481,55

t nt

10

1

( λ =

1=

1

= 0,021/

t 48,15

48,15

' !

=

(

t

=

=

)

(48,15 3)

k

t+ t

+

.

' !

3

=

(

t

=

=

)

(48,15 3)

k

t+ t

+

&

:t= 8

−

×

;

–P(8)=

e0.021= 8

t= 20

P e− ×

,0

;t= 40 P e−

×

;t= 50

–

– (20)=

0.2120=

657

–(40)= .02140=

,0 432

P

e−

×

;t= 100 P

e− ×

.

(50)=

.02150=

,035

–(100)=

0.21100= ,0

122

P

0,9
0,8
0,7				P(t)=e- t
0,6
0,5
0,4
0,3
0,2
0,1
0
0	10	20	30	40	50	60	70	80	90	100
										t, c
										t ,
	* . 2..1.							2.3

2.-4. ' 10

150, 20 .

,

t: 50, 100,.200, 300

' !2.4. ( )

=	t			=		150	= 0
	(	+	)			150 20
k	t		t			+		.

15

( λ =

1

=

1= 0, .

t

150

1

1

−

−

& t1 = 50 –P(t1) e15050 e3

,720;

=

=

=

−

1

− 10

−

1

−

20

100 Pt

;t3 =

200

,026;

t2 =

–(2) e150100e15 ,051

–P(t3) e150200e15

=

= =

=

=

=

−

1

t4= 300 –P(4t=) 150e 300= − 2= e .

2.'5. ! 6,

! : 0,81; 0,78; 0,9 0,86, : 1/10; 1/20; 1/40; 1/10; 1/40;

, !

, 1, 2, 10 20 .% ,

?

&2.5. :

λ

= λ + λ + λ + λ + λ + λ =

+

+

+

+

+

=

=

/ .

1

2

3

4

5

6101

120 1401101 501

064

-

=

1=

200= 3,1

λ

64

t .

' ! :

=

=

=

.

k

1k

2 k3 ...k 6

k0,81 0,78 0,91 0

&

t

P t−λ

t e

e

= 1

( ) =

1 =

− 0,31=

0, ;

1

1

t2= 2

–

P t −λ

2te 0,32e

t= 3

–

(

)

=

=

−

=

0, ;

2

3

Pt e−λ t e

3,03=

;

10

–

Pt e−λ t e

;

( 3)=

3=

−

,0384

t=

(

4)=

4=

−

3,04= ,042

4

Pt e−λ t

e

.

t= 20

3,05=

( 5)

=

5 =

−

,002

5

2.-6. ! -, 202

100 ,

– 4 .

! ! ,

10, 20,5030,. 40

2.6.

=

t

=

100= 0

' !

k t

+

t 100 4

.

+

= 0

' !

=

t

t

=

4

k

+

t 100 4

.

+

( λ =

1

=

1= 0 / .

t

100

16

t1= 10

– P(t)− λ 1te − 0,0110e

&

−λ

1

=

=

= 0,905;

−λ

t2e=

−

3te=

−

t2 = 20–P(2 t=)

0,0120e= 0,819t3=;30 –P(3t=)

0,0130e= 0,741t=;40

–P(t)−λ 4te −0,0140e

−λ

−

4

=

=

=

0,670t5=;50 –P(5t=)

5te=

0,0150e= 0,606.

2.-7.

! /0180

230 ,

– 8 .

! ! ,

10, 20,5030,. 40

2.7.

) λ =

=

1= 0, .

( (

1

t

230

' !

=

t

=

230=

0,

k t

+

t

230 8

.

+

= 0,

' !

=

t

t

=

8

k

+

t

230 8

.

+

–P(t)− λ 1te − 0,004310e

&

t1= 10

1 =

−λ

=

= 0,958;

t2= 20

−λ

2te=

−

–P(3t=)

3te=

−

–P(2t=)

0,0043e=20,918t3=; 30

0,0043e= 30,879;

−λ

t4e=

−

−λ

5te=

−

t4= 40 –P(4 t=)

0,0043e= 40,842t5=; 50 –P(5t=)

0,0043e= 50,806.

2.8.

,

5

( )

t1

20 2 = 22 3 = 18 4 = 25 5 = 20

= ;t

,t

,t

,t

t1

= 1;t2

=

;t

1,8 5

=

1,.2

1,5 3 = 2;

t4

=

;t

1

:

;

(

) $ ;

t=

10, 20, 3

50 ;!

k

k .

2.8.

-

2

â³äì(

. )/

+

(20

+

22 18

=

=

n

=

+

+

.

t

i=1

â³äì i

t

n

-

tðåì= (n

t.ðåì)/= in+ (1+

+1,5+

2 =1, .

=i1

( λ =

1

=

1= 0, / .

t

21

' !

=

t

=

21

=

0

k t

+

t

21 1,5

.

+

1,5= 0

' !

=

t

t

=

k

+

t

21 1,5

.

+

&

−λ

1te=

−

1t= 10 –P(1 t=)

0,04710e= 0,625;

t2 =

−λ

2te=

−

–P(3t=)

−λ

3te=

−

20 –P(2t=)

0,047e=20 0,386t3 =; 30

0,047e=300,240;

−λ

4te=

−

−λ

5te=

−

t4= 40 –P(4t=)

0,047e=400,149t5=;50 –P(5t=)

0,047e=500,093.

17

		3.		. .
: ;
; (				)				(
);		, .								–
.
		–					,
	! / , /,!				– 3/ ,
	– -			(/).					max/
" #					k!–
! p*t.									Q
" # !			=			–=			,	–
			k	/					/
				t		k			t

.

. ,

:

– Q = d , / – (

!

);

– =

/ –!

( $ $ )

Q

,

(

$

$ )%.

–

,;

–

/ –

Q

,

. =

, / .%

–

$ , ..

;

–			%	/ –				,
				,
;
–	Q		/		–			!
			,
, ;
–				/		–			,
				,
				–			=%			;
–				/ –
		Q		,
(					) #
			!
							.

# ;

*t – , t – ,

, ( , .).

18

–# ,#

,

,

Q

–

$

(

$ ),, ,

#(

) k

k

,

$ .&

ma

m , /

#

=

Q

–

Q

k Q

,# ;

– nQ,/ –

, (

)

! .

(

),

$ $ $ .

'

$ $

( , ), , , ,

# ,

! .%

# k=			Qma	, Q
			Q		m
			c
!(						–	,
) ! .
$ $						% , /		–
			$ , , ,
			,

’ : ,

, , - . “

” .(

( )

,! $ ,

! ,! .

) $ $ , $ ,

, !- , (

), - ,

( ) .

“ ” , ,

$ ,

,

.

3.1. . 3.1

m, /

(,,, ,) Q / (,,,, ) .! –

m(t).

19

)

)

)

)

)

)

)

)

" . 3.#1, m(t)

(,,, ,)

Q(t) ,(,,,)

20