24 uždavinys
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:
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* r^2* h = |
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| 1 |
| / 3 |
* r^2* h = $$\frac{1}{3}\cdot \pi\cdot r^{2}\cdot h$$ = Paaiškinimas: | 1 |
| / 3 |
* ( 6* cos(a))^2* h = $$\frac{1}{3}\cdot \pi\cdot (6\cdot cos(a))^{2}\cdot h$$ = Paaiškinimas: | 1 |
| / 3 |
* ( 6* cos(a))^2* 6* sin(a) = $$\frac{1}{3}\cdot \pi\cdot (6\cdot cos(a))^{2}\cdot 6\cdot sin(a)$$ = | 1 |
| / 3 |
* 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 |
* 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) = $$12\cdot \pi\cdot cos(a)^{2}\cdot 6\cdot sin(a)$$ =
* 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) = $$12\cdot 6\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$$ =
* cos(a)^2* sin(a) = $$12\cdot 6\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$$ = 72* π* cos(a)^2* sin(a) = $$72\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$$ =
* cos(a)^2* sin(a) = $$72\cdot \pi\cdot cos(a)^{2}\cdot sin(a)$$ = 72* π* (1- sin(a)^2)* sin(a) = $$72\cdot \pi\cdot (1-sin(a)^{2})\cdot sin(a)$$ =
* (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)) = $$72\cdot \pi\cdot (1\cdot sin(a)-sin(a)^{2}\cdot sin(a))$$ =
* ( 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)) = $$72\cdot \pi\cdot (sin(a)-sin(a)^{2}\cdot sin(a))$$ =
* (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)$$72\cdot \pi\cdot (sin(a)-sin(a)^{3})$$
* (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
* (sin(a)- sin(a)^3))′ = 0
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( 72* π* (sin(a)- sin(a)^3))′ = 0$$(72\cdot \pi\cdot (sin(a)-sin(a)^{3}))'$$ = $$0$$
* (sin(a)- sin(a)^3))′ = 0$$(72\cdot \pi\cdot (sin(a)-sin(a)^{3}))'$$ = $$0$$ 72* π* (sin(a)- sin(a)^3)′ = 0$$72\cdot \pi\cdot (sin(a)-sin(a)^{3})'$$ = $$0$$
* (sin(a)- sin(a)^3)′ = 0$$72\cdot \pi\cdot (sin(a)-sin(a)^{3})'$$ = $$0$$ Paaiškinimas: 72* π* (cos(a)- 3* sin(a)^2* sin(a)′) = 0$$72\cdot \pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot sin(a)')$$ = $$0$$
* (cos(a)- 3* sin(a)^2* sin(a)′) = 0$$72\cdot \pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot sin(a)')$$ = $$0$$ Paaiškinimas: 72* π* (cos(a)- 3* sin(a)^2* cos(a)) = 0$$72\cdot \pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot cos(a))$$ = $$0$$
* (cos(a)- 3* sin(a)^2* cos(a)) = 0$$72\cdot \pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot cos(a))$$ = $$0$$ 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$$\pi\cdot (cos(a)-3\cdot sin(a)^{2}\cdot cos(a))$$ = $$0$$(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$$cos(a)-3\cdot sin(a)^{2}\cdot cos(a)$$ = $$0$$
Paaiškinimas: cos(a)* (1- 3* sin(a)^2) = 0$$cos(a)\cdot (1-3\cdot sin(a)^{2})$$ = $$0$$
cos(a) = 0$$cos(a)$$ = $$0$$
arccos(cos(a)) = arccos(0)$$arccos(cos(a))$$ = $$arccos(0)$$
a = arccos(0)$$a$$ = $$arccos(0)$$
a =
+ π* k$$a$$ = $$\frac{\pi}{2}+\pi\cdot k$$
| π |
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* 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 |
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saknis(
) = saknis( sin(a)^2)$$\sqrt {\frac{1}{3}}$$ = $$\sqrt {sin(a)^{2}}$$
| 1 |
| / 3 |
sin(a)^2)$$\sqrt {\frac{1}{3}}$$ = $$\sqrt {sin(a)^{2}}$$saknis(
) = sin(a)$$\sqrt {\frac{1}{3}}$$ = $$sin(a)$$
| 1 |
| / 3 |
| 1 |
| / saknis(3) |
$$\frac{1}{\sqrt {3}} = \frac{\sqrt {3}}{3}$$
Sprendimas:
72* π* (sin(a)- sin(a)^3) =
* (sin(a)- sin(a)^3) = |
|
72* π* (sin(a)- sin(a)^3) = $$72\cdot \pi\cdot (sin(a)-sin(a)^{3})$$ =
* (sin(a)- sin(a)^3) = $$72\cdot \pi\cdot (sin(a)-sin(a)^{3})$$ = Paaiškinimas: 72* π* (sin(arcsin(
))- sin(arcsin(
))^3) = $$72\cdot \pi\cdot (sin(arcsin(\frac{\sqrt {3}}{3}))-sin(arcsin(\frac{\sqrt {3}}{3}))^{3})$$ =
* (sin(arcsin( | saknis(3) |
| / 3 |
| saknis(3) |
| / 3 |
72* π* (
- sin(arcsin(
))^3) = $$72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-sin(arcsin(\frac{\sqrt {3}}{3}))^{3})$$ =
* ( | saknis(3) |
| / 3 |
| saknis(3) |
| / 3 |
72* π* (
- (
)^3) = $$72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-(\frac{\sqrt {3}}{3})^{3})$$ =
* ( | saknis(3) |
| / 3 |
| saknis(3) |
| / 3 |
72* π* (
-
) = $$72\cdot \pi\cdot (\frac{\sqrt {3}}{3}-\frac{\sqrt {3}}{9})$$ =
* ( | saknis(3) |
| / 3 |
| saknis(3) |
| / 9 |
72* π* (
) = $$72\cdot \pi\cdot (\frac{2\cdot \sqrt {3}}{9})$$ =
* ( | 2* saknis(3) |
| / 9 |
| 72* π* 2* saknis(3) |
| / 9 |
| π* 72* 2* saknis(3) |
| / 9 |
π* 16* saknis(3) = $$\pi\cdot 16\cdot \sqrt {3}$$ =
* 16* saknis(3) = $$\pi\cdot 16\cdot \sqrt {3}$$ = 16* saknis(3)* π$$16\cdot \sqrt {3}\cdot \pi$$
3)* π$$16\cdot \sqrt {3}\cdot \pi$$Atsakymas: $$16\cdot \sqrt {3}\cdot \pi$$