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Sprendimas:

Kūgio pagrindo spindulys r, sudaromoji L = 6.

$\frac{r}{L}$ = $cos(a)$

$r$ = $cos(a)\cdot L$ = $6\cdot cos(a)$ (1)

kai $a$ = $\frac{\pi}{3}$

$r$ = $6\cdot cos(\frac{\pi}{3})$ = $\frac{1}{2}\cdot 6$ = $3$

Šoninio paviršiaus plotas $\pi\cdot r\cdot L$ = $\pi\cdot 3\cdot 6$ = $18\cdot \pi$

Atsakymas: 18π

Sprendimas:

Kūgio aukštinė h.

$\frac{h}{L}$ = $sin(a)$

$h$ = $sin(a)\cdot L$ = $sin(a)\cdot 6$ = $6\cdot sin(a)$ (2)

Kūgio tūrio formulė $V$ = $\frac{1}{3}\cdot \pi\cdot r^{2}\cdot h$

Į ją statome (1) ir (2) gautas išraiškas:

1

/ 3

* π

* r^²* h =

1

/ 3

* π

* r^²* h = $\frac{1}{3}\cdot \pi\cdot r^{2}\cdot h$ =

1

/ 3

* π

* 36* cos(a)^²* 6* sin(a) = $\frac{1}{3}\cdot \pi\cdot 36\cdot cos(a)^{2}\cdot 6\cdot sin(a)$ =

1

/ 3

* 36* π

* cos(a)^²* 6* sin(a) = $\frac{1}{3}\cdot 36\cdot \pi\cdot cos(a)^{2}\cdot 6\cdot sin(a)$ =

12* π

* cos(a)^²* 6* sin(a) = $12\cdot \pi\cdot cos(a)^{2}\cdot 6\cdot sin(a)$ =

12* 6* π

* cos(a)^²* sin(a) = $12\cdot 6\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$ =

72* π

* cos(a)^²* sin(a) = $72\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$ =

72* π

* (1- sin(a)^²)* sin(a) = $72\cdot \pi\cdot (1-sin(a)^{2})\cdot sin(a)$ =

72* π

* ( 1* sin(a)- sin(a)^²* sin(a)) = $72\cdot \pi\cdot (1\cdot sin(a)-sin(a)^{2}\cdot sin(a))$ =

72* π

* (sin(a)- sin(a)^²* sin(a)) = $72\cdot \pi\cdot (sin(a)-sin(a)^{2}\cdot sin(a))$ =

72* π

* (sin(a)- sin(a)^³) $72\cdot \pi\cdot (sin(a)-sin(a)^{3})$

Sprendimas:

Tūrio išvestinę prilyginame nuliui:

( 72* π

* (sin(a)- sin(a)^³))′ =
0

( 72* π

* (sin(a)- sin(a)^³))′ = 0 $(72\cdot \pi\cdot (sin(a)-sin(a)^{3}))'$ = 0

72* π

* (sin(a)- sin(a)^³)′ = 0 $72\cdot \pi\cdot (sin(a)-sin(a)^{3})'$ = 0

π

* (cos(a)- 3* sin(a)^²* cos(a)) = 0 $\pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot cos(a))$ = 0

(cos(a)- 3* sin(a)^²* cos(a)) = 0 $(cos(a)-3\cdot sin(a)^{2}\cdot cos(a))$ = 0

cos(a)- 3* sin(a)^²* cos(a) = 0 $cos(a)-3\cdot sin(a)^{2}\cdot cos(a)$ = 0

cos(a) = 0 $cos(a)$ = 0

arccos(cos(a)) = arccos(0) $arccos(cos(a))$ = $arccos(0)$

a = arccos(0) $a$ = $arccos(0)$

a =

π

/ 2

+ π

* k $a$ = $\frac{\pi}{2}+\pi\cdot k$

1- 3* sin(a)^² = 0 $1-3\cdot sin(a)^{2}$ = 0

1 = 3* sin(a)^²+0 $1$ = $3\cdot sin(a)^{2}+0$

1 = 3* sin(a)^² $1$ = $3\cdot sin(a)^{2}$

1

/ 3

= sin(a)^² $\frac{1}{3}$ = $sin(a)^{2}$

saknis(

1

/ 3

) = saknis(

sin(a)^²) $\sqrt {\frac{1}{3}}$ = $\sqrt {sin(a)^{2}}$

saknis(

1

/ 3

) = sin(a) $\sqrt {\frac{1}{3}}$ = $sin(a)$

1

/ saknis(

3)

= sin(a) $\frac{1}{\sqrt {3}}$ = $sin(a)$

$\frac{1}{\sqrt {3}}$ = $\frac{\sqrt {3}}{3}$

Sprendimas:

72* π

* (sin(a)- sin(a)^³) =

72* π

* (sin(a)- sin(a)^³) = $72\cdot \pi\cdot (sin(a)-sin(a)^{3})$ =

72* π

* (

saknis(

3)

/ 3

- sin(arcsin(

saknis(

3)

/ 3

))^³) = $72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-sin(arcsin(\frac{\sqrt {3}}{3}))^{3})$ =

72* π

* (

saknis(

3)

/ 3

- (

saknis(

3)

/ 3

)^³) = $72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-(\frac{\sqrt {3}}{3})^{3})$ =

72* π

* (

saknis(

3)

/ 3

-

saknis(

3)

/ 9

) = $72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-\frac{\sqrt {3}}{9})$ =

72* π

* (

2* saknis(

3)

/ 9

) = $72\cdot \pi\cdot (\frac{2\cdot \sqrt {3}}{9})$ =

72* π

* 2* saknis(

3)

/ 9

= $\frac{72\cdot \pi\cdot 2\cdot \sqrt {3}}{9}$ =

π

* 72* 2* saknis(

3)

/ 9

= $\frac{\pi\cdot 72\cdot 2\cdot \sqrt {3}}{9}$ =

π

* 16* saknis(

3) = $\pi\cdot 16\cdot \sqrt {3}$ =

16* saknis(

3)* π

$16\cdot \sqrt {3}\cdot \pi$

Atsakymas: $16\cdot \sqrt {3}\cdot \pi$

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