Silindr, konus va shar: matematika darsi

Aylanish jismlari kombinatsiyasi o'q kesimida tekis masalaga aylanadi: konusga ichki chizilgan shar o'q kesimi uchburchagiga ichki chizilgan aylana bilan bir xil. Radiusi $9$ , balandligi $12$ konusda $r=\dfrac{S}{p}=\dfrac{108}{24}=\dfrac{9}{2}$ . Bu darsda barcha asosiy juftliklar, formulalar va eng katta hajm masalalari beriladi.

Silindrga ichki chizilgan shar

Silindrga shar faqat o'q kesimi kvadrat bo'lganda ichki chiziladi: $H=2R$ , shar radiusi $R$ .

O'q kesimi: kvadrat va unga ichki chizilgan aylana

Kattalik	Silindr	Shar	Nisbat
KattalikHajm	Silindr $2\pi R^3$	Shar $\dfrac{4}{3}\pi R^3$	Nisbat $\dfrac{V_{\text{shar}}}{V_{\text{sil}}}=\dfrac{2}{3}$
KattalikTo'la sirt	Silindr $6\pi R^2$	Shar $4\pi R^2$	Nisbat $\dfrac{2}{3}$
KattalikYon sirt va sfera	Silindr $4\pi R^2$	Shar $4\pi R^2$	Nisbat $1$

Shart $H=2R$ bajarilmasa, silindr ichidagi eng katta shar radiusi $\min\left(R,\dfrac{H}{2}\right)$ . Radiusi $6$ , balandligi $8$ silindrdan yo'nilgan eng katta shar radiusi $4$ , u yon sirtga urinmaydi. $\pi\approx 3$ da chiqindi:

3\cdot 36\cdot 8-\frac43\cdot 3\cdot 64

=864-256=608

Sharga ichki chizilgan silindr

O'q kesimi to'g'ri to'rtburchak, uning diagonali shar diametri. Bu yerda shar radiusi $R$ , silindr radiusi $r$ .

R^2=r^2+\left(\frac{H}{2}\right)^2

Sharga istalgan silindr ichki chiziladi

R=13

H=10

r=12

Shart	Natija	Qiymat
ShartO'q kesimi kvadrat	Natija $r=\dfrac{R}{\sqrt2}$ , $H=R\sqrt2$	Qiymat $V=\dfrac{\pi R^3}{\sqrt2}$
ShartHajm eng katta	Natija $H=\dfrac{2R}{\sqrt3}$ , $r^2=\dfrac{2R^2}{3}$	Qiymat $V_{\max}=\dfrac{4\pi R^3}{3\sqrt3}$
ShartYon sirt eng katta	Natija $r=\dfrac{H}{2}=\dfrac{R}{\sqrt2}$	Qiymat $S_{\max}=2\pi R^2$

Eng katta hajm, bu yerda ${0<H<2R}$ :

V=\pi H\left(R^2-\frac{H^2}{4}\right)

V'(H)=\pi\left(R^2-\frac{3H^2}{4}\right)=0

Eng katta yon sirt:

r\cdot\frac{H}{2}\le\frac{1}{2}\left(r^2+\frac{H^2}{4}\right)

=\frac{R^2}{2}

Tenglik

{r=\dfrac{H}{2}}

Yarim sharga ichki chizilgan silindr. Pastki asos yarim shar asosida yotadi, shuning uchun to'liq $H$ turadi, ${\dfrac{H}{2}}$ emas:

R^2=r^2+H^2

Yarim shar

H=\frac{R}{\sqrt3}

r^2=\frac{2R^2}{3}

Yarim shardagi eng katta hajmli silindr

Konusga ichki chizilgan shar

Shar konus asosiga va yon sirtiga urinadi, markazi balandlikda. O'q kesimida uchburchakka ichki chizilgan aylana. $R$ konus radiusi, $\rho$ shar radiusi.

\rho=\frac{RH}{R+l}

\rho=R\operatorname{tg}\frac{\alpha}{2}

\dfrac{S}{p}

formulasi:

S=RH

p=R+l

\alpha

yasovchi va asos orasidagi burchak

T

urinish nuqtasi,

KT

urinish aylanasi radiusi

Yasovchi va balandlik orasidagi $\beta$ burchak orqali: $SI=\dfrac{\rho}{\sin\beta}$ , ya'ni $H=\rho+\dfrac{\rho}{\sin\beta}$ .

O'q kesimi	$\rho$
To'g'ri burchakli uchburchak, $H=R$	$R(\sqrt2-1)$
Muntazam uchburchak, $H=R\sqrt3$	$\dfrac{H}{3}=\dfrac{R}{\sqrt3}$
$l=3R$ , $H=2\sqrt2\,R$	$\dfrac{H}{4}$

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