(Upr. 20.09. 15:38) | Nahlásit

Matematika DÚ

Témata: Nezařazené
Diskuze
(Upr. 20.09. 16:29) | Nahlásit
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Vypočítejte výšku akvária ve tvaru kvádru, známe-li velikost základy a=30 cm, b=50 cm a víme, že má akvárium 60 litrů objem.

a=30 cm = 3 dm
b=50 cm = 5 dm
c=? dm
V=60 litrů = 60 dm3
V=a*b*c
60=3*5*c
c=60/15=4 dm
===

-------------------------------------------------------------------------------

Mistr s učněm mají vykonat práci.
Mistr ji udělá za 6 dní.
Mistr ji udělá za 10 dní.

v=V/t; rychlost práce

v(M)=V/6
v(U)=V/10

V(M) + V(U) = V
v(M)*T + v(u)*T = V
V/6*T + V/10*T = V
1/6*T + 1/10*T = 1
(1/6 + 1/10)*T = 1
(10/60 + 6/60)*T = 1
(16/60*T = 1
T = 60/16=3,75 dní
===

-------------------------------------------------------------------------------

Vodní nádrž se 1. přívodem naplní za 36 minut.
Vodní nádrž se 2. přívodem naplní za 45 minut.

za jak dlouho se naplní nádrž, když se 9 minut plní 1. přívodem a pak oběma přívody ?

v(1)=V/36
v(2)=V/45

v=V/t; rychlost plnění

V0+V1+V2=V
v(1)*9+v(1)*t + v(2)*t=V
V/36*9+V/36*t + V/45*t=V
1/36*9+1/36*t + 1/45*t=1
1/4+1/36*t + 1/45*t=1
1/36*t + 1/45*t=1-1/4
(1/36 + 1/45)*t=3/4
(1/36 *45/45 + 1/45 * 36/36)*t=3/4
(1/36 *45/45 + 1/45 * 36/36)*t=3/4

(45+36)/1620*t = 3/4
t = 3/4*1620/(45+36)=15 minut

T=t+9=24 minut
==============

-------------------------------------------------------------------------------

V Kocourkově se 1x ročně plní sýpka.
Kočky ji zaplní za 2 hodiny.
Kocouři ji zaplní za 5 hodin.
Myšky ji vyprázdní za 10 hodin.

Za jak dlouho by sýpku naplnili všichni najednou ?

v(k)=V/2
v(K)=V/5
v(m)=V/10

V(k)+V(K)+V(m)=V
V/2*T+V/5*T+V/10*T=V
1/2*T+1/5*T+1/10*T=1
(1/2+1/5+1/10)*T=1
(5/10+2/10+1/10)*T=1
(8/10)*T=1

T=10/8=1,25 hod = 1 hod 15 minut.
===
-------------------------------------------------------------------------------
(Upr. 02.10. 15:11) | Nahlásit
1)

( 15^(1/3) * 27^(-1/2) )^(-3) / ( 25^(1/4) * 9^(1/8) )^(-2) * ( 3 * 27^(1/4) )^(1/3) / ( 9^(1/3) )^(1/2) =
( 25^(1/4) * 9^(1/8) )^2 / ( 15^(1/3) * 27^(-1/2) )^3 * ( 3 * 27^(1/4) )^(1/3) / ( 9^(1/3) )^(1/2) =
( 5²^(2/4) * 9^(2/8) ) / ( 15^(3/3) * 27^(-3/2) ) * ( 3^(1/3) * 27^(1/12) ) / ( 9^(1/6) ) =
( 5 * (3²)^(1/4) ) / ( 15^(3/3) * (3³)^(-3/2) ) * ( 3^(1/3) * (3³)^(1/12) ) / ( (3²)^(1/6) ) =
( 5 * 3^(2/4) ) / ( 15 * (3)^(-9/2) ) * ( 3^(1/3) * (3)^(3/12) ) / ( (3)^(2/6) ) =
( 5 * 3^(1/2) ) / ( 3*5 * (3)^(-9/2) ) * ( 3^(1/3) * (3)^(1/4) ) / ( (3)^(1/3) ) =
( 5 * 3^(1/2 - 1 + 9/2) ) / ( 5 ) * ( 3^(1/3 + 1/4 - 1/3) ) =
( 3^(1/2 - 1 + 9/2) ) * ( 3^(1/3 + 1/4 - 1/3) ) =
3^(1/2 - 1 + 9/2 + 1/3 + 1/4 - 1/3) =
3^(17/4) = 3^(4 + 1/4) = 3^(3 + 1 + 1/4) =

= 3^3 * 3^(5/4) = 27 * 3^(5/4)
===

2)

( a^(5) * b^(1/2) * a^(-1/4) )^(1/3) / ( a² * (a*b³)^(1/5) )^2 =
( a^(5 - 1/4) * b^(1/2) )^(1/3) / ( a² * a^(1/5) * b^(3/5) )^2 =
( a^(20/4 - 1/4) * b^(1/2) )^(1/3) / ( a^4 * a^(2/5) * b^(6/5) ) =
( a^(1/3 * 19/4) * b^(1/6) ) / ( a^(4 + 2/5) * b^(6/5) ) =
( a^(19/12) * b^(1/6) ) / ( a^(20/5 + 2/5) * b^(6/5) ) =
( a^(19/12) * b^(1/6) ) / ( a^(22/5) * b^(6/5) ) =
( a^(19/12 - 22/5) * b^(1/6 - 6/5) ) =
( a^(19/12 * 5/5 - 22/5 * 12/12) * b^(5/30 - 36/30) ) =
( a^(5*19/60 - 12*22/60) * b^(-31/30) ) =
( a^(95/60 - 264/60) * b^(-31/30) ) =
( a^(-169/60) * b^(-31/30) ) =

= 1 / ( a^(169/60) * b^(31/30) )
================================

3)

( ( a^(4/3) )^(1/5) )^(3/2) / ( (a^4)^(1/5) )^3 * ( (a * (a² * b)^(1/3) )^(1/2) )^4 / ( ( a * b^(1/2) )^(1/3) )^6 =
a^(4/3 * 3/10) / a^(12/5) * (a * (a² * b)^(1/3) )^2 / ( a * b^(1/2) )^(2) =
a^(4/10) / a^(12/5) * a^(2) * (a² * b)^(2/3) / ( a^(2) * b^(2/2) ) =
a^(2/5) / a^(12/5) * a^(2) * a^(4/3) * b^(2/3) / ( a^2 * b ) =
a^(2/5 - 12/5 + 2 + 4/3 - 2) * b^(2/3 - 1) =
a^(-30/15 + 20/15) * b^(-1/3) =
a^(-10/15) * b^(-1/3) =
a^(-2/3) * b^(-1/3) =

= 1 / ( a^2 * b )^(1/3)
= 1 / ( a^(2/3) * b^(1/3) )
=========================

4)

( 2^(1/2) / (1-x²)^(-1) + 2^(3/2) / x^(-2) ) * x^(-2)/( 1+x^(-2) ) =
( 2^(1/2) * (1-x²) + 2^(3/2) * x² ) * 1/( x² + 1) =
( 2^(1/2) - 2^(1/2) * x² + 2^(3/2) * x² ) * 1/( x² + 1) =
( 2^(1/2) - 2^(1/2) * x² + 2^(1+1/2) * x² ) * 1/( x² + 1) =
( 2^(1/2) - 2^(1/2) * x² + 2*2^(1/2) * x² ) * 1/( x² + 1) =
2^(1/2)*( 1 - x² + 2*x² ) * 1/( x² + 1) =
2^(1/2)*( 1 + x² ) * 1/( x² + 1) =

= 2^(1/2) = √2
==============

5)

( 1/(a - √2) - (a² + 4)/( a³ - (√2)³ ) ) * ( a/√2 + 1 + √2/a ) =
( 1/(a - √2) - (a² + 4)/( a³ - (√2)³ ) ) * ( a² + a*√2 + 2 ) * 1/√2 * 1/a=

vzorec: a³ - b³ = (a-b)*(a²+ab+b²)

( (a²+a*√2+(√2)²)/(a²+a*√2+(√2)²)/(a - √2) - (a² + 4)/( a³ - (√2)³ ) ) * ( a² + a*√2 + 2 ) * 1/√2 * 1/a=
( (a²+a*√2+2)/( a³ - (√2)³ ) - (a² + 4)/( a³ - (√2)³ ) ) * ( a² + a*√2 + 2 ) * 1/√2 * 1/a=
(a*√2 - 2)/( a³ - (√2)³ ) * ( a² + a*√2 + 2 ) * 1/√2 * 1/a=
(a*√2*1/√2 - 2/√2)/( a³ - (√2)³ ) * ( a² + a*√2 + 2 ) * 1/a=
(a - √2)/( a³ - (√2)³ ) * ( a² + a*√2 + 2 ) * 1/a=

= 1/a
=====

6)

( (6/10)^0 - (1/10)^(-1) ) / ((3/2³)^(-1) * (3/2)^(3) * (-1/3)^(-1) )
( 1 - 10 ) / ( 2³/3 * (3/2)^(3) * (-3) ) =
( - 9 ) / ( 2³ * (3/2)^(3) * (-1) ) =
9 / ( 2³ * (3/2)^(3) ) =
3² / ( 2³ * (3/2)^(3) ) =
3^(2-3)/ ( 2³ * (1/2)^(3) ) =
3^(-1)/ 1 =

= 1/3
=====

7)

((√3 + √11)/(√3 - √11))² + ((√11 - √3)/(√11 + √3))² =
( (√3 + √11)/(√3 - √11) * (√3 + √11)/(√3 + √11) )² + ( (√11 - √3)/(√11 + √3) * (√11 - √3)/(√11 - √3) )² =
( (√3 + √11)²/(3 - 11) )² + ( (√11 - √3)²/(11 - 3) )² =
( (3 + 2*√3√11 + 11)/(-8) )² + ( (11 - 2*√11*√3 + 3)/(8) )² =
( (7 + √3√11)/(-4) )² + ( (7 - √11*√3)/(4) )² =
( (7 + √33)/(-4) )² + ( (7 - √33)/(4) )² =
(49 + 2*7*√33 + 33)/(16) + (49 - 2*7*√33 + 33)/(16) =
(49 + 33)/(16) + (49 + 33)/(16) =
2*(49 + 33)/(16) =
2*(82)/(16) =

= (41)/4
========
(Upr. 13.10. 13:07) | Nahlásit
1)

(a²/b² - a/b) / ( (a² + b²)/ab - 2 ) / (a²/ b) =
(a²/b² - a/b * b/b) / ( (a² + b²)/ab - 2 * ab/ab ) * b/a² =
(a²/b² - ab/b²) / ( (a² - 2ab + b²)/ab) * b/a² =
(a² - ab)/b / ( (a² - 2ab + b²)/ab) * 1/a² =
(a² - ab)/b * ab/( (a² - 2ab + b²)) * 1/a² =
(a² - ab) * 1/(a² - 2ab + b²) * 1/a =
a*(a - b) * 1/(a-b)² * 1/a =
a * 1/(a-b) * 1/a =

= 1/(a-b)
===

2)

√( 3/5 * ³√(3/5 * √(5/3) ) ) =
( 3/5 * (3/5 * √(5/3) )^(1/3) )^(1/2) =
( 3/5 * (3/5 * (5/3)^(1/2) )^(1/3) )^(1/2) =

((3/5)^(1/2) * (3/5 * (5/3)^(1/2) )^(1/6) ) =

(3/5)^(1/2) * (3/5)^(1/6) * (3/5)^(-1/12) =

(3/5)^(1/2 + 1/6 - 1/12) =
(3/5)^(6/12 + 2/12 - 1/12) =

= (3/5)^(7/12)
===

3)

( 2x/(x+y) + y/(x-y) + y²/(y²-x²) ) / ( 1/(x-y) + x/(x²-y²) ) =
( 2x/(x+y) * (x-y)/(x-y) + y/(x-y) * (x+y)/(x+y) - y²/(x²-y²) ) / ( 1/(x-y) * (x+y)/(x+y) + x/(x²-y²) ) =
( 2x/(x²-y²) * (x-y) + y/(x²-y²) * (x+y) - y²/(x²-y²) ) / ( 1/(x²-y²) * (x+y) + x/(x²-y²) ) =
( 2x*(x-y)/(x²-y²) + y*(x+y)/(x²-y²) - y²/(x²-y²) ) / ( (x+y)/(x²-y²) + x/(x²-y²) ) =
( (2x²-2xy)/(x²-y²) + (xy+y²)/(x²-y²) - y²/(x²-y²) ) * (x²-y²)/(2x+y) =
( (2x²-2xy) + (xy+y²) - y² ) /(2x+y) =

= (2x²-xy) /(2x+y)
= x*(2x-y) /(2x+y)
===

4)

√3(√3+√5) - √5(√3+√5) =
3+√(3*5) - √(3*5) - 5 =

3-5 = -2
===
(Upr. 08.11. 16:05) | Nahlásit
Př.

1. (x² + 1/x) / (x + 1/x -1) =
V: = 1+x

2. ( (a+b)/(a-b) - 1 ) / ( (a+b)/(a-b) + 1 ) =
V: = b/a


1. |x+1| + |x-2| = 3
2. 3x + |6x+5| = 1
3. |x-1| = 5

---

Př 1.

NB:
x+1 = 0
x = -1

x-2 = 0
x = 2


1) I1: (-∞,-1>

(x+1) , (x-2)
(-) , (-)

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

2) I2: <-1,+2>

(x+1) , (x-2)
(+) , (-)

|x+1| + |x-2| = 3
(x+1) + -(x-2) = 3
3 = 3; v tomto intervalu vyhovují všechna x
---

3) I3: <+2,+∞)

(x+1) , (x-2)
(+) , (+)

|x+1| + |x-2| = 3
(x+1) + (x-2) = 3
2x = 4
x = +2
---

K={x:<-1,+2>}

https://www.priklady.eu/cs/matematika/rovnice-s-absolutni-hodnotou.alej
(Upr. 15.11. 13:05) | Nahlásit
1. 5x*(x+4) / ( (x-4)*(2x-3) ) <=0
2. (2-x)*(x+1) / (x+7) < 0
3. (5x-2)*(x+7)*(x+3) / ( (x-7)*(x-3) ) <=0

1)

5x*(x+4) / ( (x-4)*(2x-3) ) <=0

D: x ∈ R - {+4,+3/2}

NB:
(x-0)=0; x=0
(x+4)=0; x=-4
(x-4)=0; x=+4
(2x-3)=0; x=+3/2

I1: (-∞,-4)
I2: (-4,0)
I3: (0,+3/2)
I4: (+3/2,+4)
I5: (+4,+∞)

I1: (-∞,-4); volím: -5
5x ; (x+4) ; (x-4) ; (2x-3)
- | - | - | - | (+)

I2: (-4,0); volím: -1
5x ; (x+4) ; (x-4) ; (2x-3)
- | + | - | - | (-)

...


2)

(2-x)*(x+1) / (x+7) < 0

D(f) = R \ {-7}
D: x ∈ R - {-7}

NB:
(2-x)=0;x=+2
(x+1)=0;x=-1
(x+7)=0;x=-7

3)

(5x-2)*(x+7)*(x+3) / ( (x-7)*(x-3) ) <=0
(Upr. 15.11. 13:05) | Nahlásit
.
(Upr. 21.11. 10:46) | Nahlásit
1)

[ (1-x) / (1-x+x²) + (1+x) / (1+x+x²) ] / [ (1+x) / (1+x+x²) - (1-x) / (1-x+x²) ] =
[ (1-x)*(1+x+x²) + (1+x)*(1-x+x²) / (1-x+x²)*(1+x+x²) ] / [ (1+x)*(1-x+x²) - (1-x)*(1+x+x²) / (1+x+x²)*(1-x+x²) ] =
[ (1-x)*(1+x+x²) + (1+x)*(1-x+x²) ] / [ (1+x)*(1-x+x²) - (1-x)*(1+x+x²) ] =
[ ((1-x)+(1-x)*x+(1-x)*x²) + ((1+x)-(1+x)*x+(1+x)*x²) ] / [ ((1+x)-(1+x)*x+(1+x)*x²) - ((1-x)+(1-x)*x+(1-x)*x²) ] =
=[ 1+x² ] / [ x*x² ] =

[ 2 ] / [ 2*x*x² ] =

= 1 / [ x*x² ] = 1/x³
===

https://www.wolframalpha.com/input?i=%5B+%281-x%29+%2F+%281-x%2Bx%C2%B2%29+%2B+%281%2Bx%29+%2F+%281%2Bx%2Bx%C2%B2%29+%5D+%2F+%5B+%281%2Bx%29+%2F+%281%2Bx%2Bx%C2%B2%29+-+%281-x%29+%2F+%281-x%2Bx%C2%B2%29+%5D+%3D

2)

[ (a+b)/(a-b) - 1 ] / [ (a+b)/(a-b) + 1 ] =
[ (a+b)/(a-b) - (a-b)/(a-b) ] / [ (a+b)/(a-b) + (a-b)/(a-b) ] =
[ (2b)/(a-b) ] / [ (2a)/(a-b) ] =
= 2b/2a = b/a
===

3)

(x² + 1/x) / (x + 1/x - 1) =
(x*x² + 1)/x / (x*x + 1 - x)/x =
(x*x² + 1) / (x*x - x + 1) =
(x+1)*(x²-x+1) / (x*x - x + 1) =
=x+1
====

?)

x=?

x = 5 (mod 18)
x = 5 (mod 21)
x = 5 (mod 24)

NSN: LCM(18,21,24)=504

x=504*n + 5
===========


?)

(√(2a) - 2a/(a+√(2a))) : (√(2a) - 2)/(a-2) =
(√(2a) - 2a/(a+√(2a))) * (a-2)/(√(2a) - 2) =
(√(2a)*(a+√(2a)) - 2a)/(a+√(2a)) * (a-2)/(√(2a) - 2) =
(√(2a)*a+√(2a)*√(2a) - 2a)/(a+√(2a)) * (a-2)/(√(2a) - 2) =

(√(2a)*a+2a - 2a)/(a+√(2a)) * (a-2)/(√(2a) - 2) =
(√(2a)*a)/(a+√(2a)) * (a-2)/(√(2a) - 2) * (√(2a) + 2)/(√(2a) + 2) =
(√(2a)*a)/(a+√(2a)) * (a-2)/((2a) - 4) * (√(2a) + 2) =
(√(2a)*a)/(a+√(2a)) * 1/2 * (√(2a) + 2) =
a/(a+√(2a)) * 1/2 * (2a + 2*√(2a)) =
a/(a+√(2a)) * (a + √(2a)) = a

?) [?]

7) [4]

⁵√( ( a^(1/2)*a^(-1) / (³√a) ) ^ (-3) ) =
⁵√( ( a^(1/2 - 1 - 1/3) ) ^ (-3) ) =
⁵√( ( a^(3/6 - 6/6 - 2/6) ) ^ (-3) ) =
⁵√( ( a^(-5/6) ) ^ (-3) ) =
⁵√( ( a^( (-3) * (-5/6) ) ) =
⁵√( a^( 5/2 ) ) =
= a^( 1/2 ) = √a
===

8) [3]

(10^(1/3)*8^(-1/2))^(-3) / (25^(1/4)*4^(1/8))^(-2) : √(2*³√4)/³√(2*⁴√8) =
(25^(1/4)*4^(1/8))^2 /(10^(1/3)*8^(-1/2))^3 : √(2*³√4)/³√(2*⁴√8) =
A / B =

A =
(25^(1/4)*4^(1/8))^2 /(10^(1/3)*8^(-1/2))^3 =
(2^(1/2)) /(2 * 2^(-9/2)) =
=16=2^4
-------
√(2*³√4) / ³√(2*⁴√8) =
(2*4^(1/3))^(1/2) / (2*8^(1/4))^(1/3) =
2^(5/6) / (2^(1+3/4))^(1/3) =
2^(5/6) / 2^(7/12) =
2^(10/12 - 7/12) = 2^(3/12) =
=2^(1/4)
--------

= A/B = 2^4 / 2^(1/4) = 2^(4 - 1/4) = 2^(16/4 - 1/4) = 2^(15/4)

= 2^(15/4)
===
c) je správně

9) [18]

(4 - 2/(√x + 1))*(1 + √x/(√x - 1)) - 6/(x-1) =

=8
===

10) [9]

2*a^(1/3)/(a^(4/3)-3*a^(1/3)) - a^(2/3)/(a^(5/3)-a^(2/3)) - (a+1)/(a^2-4a+3) =
2*a^(1/3)/(a^(3/3+1/3)-3*a^(1/3)) - a^(2/3)/(a^(3/3+2/3)-a^(2/3)) - (a+1)/(a^2-4a+3) =
2/(a^(3/3)-3) - 1/(a^(3/3)-1) - (a+1)/(a^2-4a+3) =
2/(a-3) - 1/(a-1) - (a+1)/(a^2-4a+3)=
(2*(a-1) - (a-3) )/ (a-3)*(a-1) - (a+1)/(a^2-4a+3)=
((a+1) )/ (a-3)*(a-1) - (a+1)/(a^2-4a+3)=
((a+1) )/ (a^2-4a+3) - (a+1)/(a^2-4a+3)=

=0
===
d) je správně
(Upr. 21.11. 10:34) | Nahlásit
VZORCE:

(a+b)²=a²+2ab+b²
(a-b)²=a²-2ab+b²

a²-b²=(a-b)*(a+b)

a³-b³=(a-b)*(a²+ab+b²)
a³+b³=(a+b)*(a²-ab+b²)
(Upr. dnes 15:54) | Nahlásit
1) -3x > 6; v oboru R - reálná čísla

-3x > 6
+3x = +3x
=--------
0 > 6 + 3x
-6 > 3x
-2 > x
x < -2
======

x=(-∞,-2); správně je: a) b)
==========

2) -5x>=-1; v oboru N - přirozená čísla

-5x>=-1
+5x=+5x
+1=+1
=------
1>=5x; :5 (vydělíme pěti)
1/5>=x
x<=1/5
======

N={0,1,2,3, ...}; správně je: d)

K={0}
=====

N={1,2,3, ...}; správně je: a)

K={}=0=∅
=====

https://cs.wikipedia.org/wiki/Pr%C3%A1zdn%C3%A1_mno%C5%BEina

3) 6-2x <= 3x-4; v intervalu (0,3)

6-2x <= 3x-4
10 <= 5x
10/5 <= x
x >= 2
======

K=<2,3)
=======

4) 2x-5< 4-x; největší číslo, které je řešením v oboru: Z - celá čísla

2x-5< 4-x
3x<9
x < 9/3
x < 3
=====

x=2; správně je: c)
=====

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