3
Shar va sfera
maktab darsligi: matematika 11-sinf, aylanish jismlari
Maqsad: Shar hajmi, sfera yuzi, kesim va chizilgan jismlar masalalarini yechish.
Qoida va g‘oya
Shar hajmi V = 4 pi R^3/3, sfera sirti S = 4 pi R^2. Shar tekislik bilan kesilsa, kesim doira. Markazdan tekislikkacha masofa d bo‘lsa, kesim radiusi r = sqrt(R^2 - d^2). Kub ichiga chizilgan sharning radiusi qirraning yarmiga, kubga tashqi chizilgan sharning diametri kub diagonaliga (a sqrt(3)) teng. Silindrga chizilgan sharda h = 2R = 2r. Silindrga tashqi chizilgan shar uchun R^2 = r^2 + (h/2)^2. Ikki sharning radiuslari nisbati k bo‘lsa, sirtlar k^2, hajmlar k^3 nisbatda.
Namunaviy masalalar
Radiusi 3 bo‘lgan shar sirti va hajmini toping.
Yechim: S = 4 pi · 9 = 36 pi. V = 4 pi · 27/3 = 36 pi.
Javob: S = 36 pi, V = 36 pi
Yechim: S = 4 pi · 9 = 36 pi. V = 4 pi · 27/3 = 36 pi.
Javob: S = 36 pi, V = 36 pi
Radiusi 5 bo‘lgan shar markazidan 3 masofada tekislik bilan kesildi. Kesim yuzini toping.
Yechim: r = sqrt(25 - 9) = 4. S = 16 pi.
Javob: 16 pi
Yechim: r = sqrt(25 - 9) = 4. S = 16 pi.
Javob: 16 pi
Sfera sirti 100 pi. Shar hajmini toping.
Yechim: 4 pi R^2 = 100 pi, R = 5. V = 4 pi · 125/3 = 500 pi/3.
Javob: 500 pi/3
Yechim: 4 pi R^2 = 100 pi, R = 5. V = 4 pi · 125/3 = 500 pi/3.
Javob: 500 pi/3
Tipik xatolar
- Hajm va sirt formulalarini aralashtirish: R^3 va R^2.
- Kesim radiusi uchun r = R - d deb yozish.
- Kub diagonalini qirraga teng deb olish.
Mashq
1
Radiusi 6 bo‘lgan shar hajmi V = k pi bo‘lsa, k ni toping.
288
2
Ikki sharning radiuslari 2 va 6. Hajmlari nisbatini toping.
27
3
Radiusi 4 bo‘lgan shar silindrga ichki chizilgan. Silindr hajmi k pi bo‘lsa, k ni toping.
128
4
Qirrasi 4 bo‘lgan kubga tashqi chizilgan shar sirti k pi bo‘lsa, k ni toping.
48