Mafunso ndi Zitsanzo za Ma Circle Sectors
Magawo a bwalo ndi mutu wofunikira kwambiri mu masamu womwe umapezeka kawirikawiri m'mayeso ndi mafunso ochita masewera olimbitsa thupi. Gawo ndi gawo la bwalo lozunguliridwa ndi ma radii awiri ndi mzere wolumikiza. M'nkhaniyi, tikambirana zitsanzo zingapo za mavuto pa magawo a bwalo, pamodzi ndi kufotokozera mwatsatanetsatane, kuti timvetsetse bwino.
Tanthauzo la Gawo la Mzere
Gawo la bwalo ndi gawo la bwalo lozunguliridwa ndi ma radii awiri ndi arc imodzi. Dera la gawo limawerengedwa kutengera gawo la dera lonse la bwalo. Fomula yayikulu yomwe imagwiritsidwa ntchito kuwerengera gawo ndi iyi:
– Malo Ochitira Milandu: \[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
– Kutalika kwa Arc : \[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
Kumene:
– \( \theta \) ndi kukula kwa ngodya ya gawo mu madigiri,
– \( r \) ndi utali wa bwalo,
– \( \pi \) ndi chinthu chosasintha (pafupifupi 3.14159).
Mafunso ndi Kukambirana Zitsanzo
Funso 1:
Mwapatsidwa bwalo lozungulira la 10 cm ndi gawo lokhala ndi ngodya yapakati ya 90°. Werengani dera la gawolo.
Kukambirana:
Ndizodziwika kuti:
– \( r = 10 \) cm
– \( \theta = 90^\circ \)
Timagwiritsa ntchito njira yofotokozera gawo la gawo:
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[L_juring = \frac{90^\circ}{360^\circ} \times \pi \times (10\text{ cm})^2\]
\[L_juring = \frac{1}{4} \times \pi \times 100\text{ cm}^2\]
\[L_juring = 25\pi\text{ cm}^2\]
Ngati \( \pi \) yatengedwa kukhala 3.14, ndiye kuti:
\[L_juring = 25 \nthawi 3.14\malemba{ cm}^2 = 78.5\malemba{ cm}^2\]
Chifukwa chake, dera la gawoli ndi 78.5 cm².
Funso 2:
Gawo la bwalo lili ndi utali wa masentimita 7 ndi kutalika kwa arc wa masentimita 11. Dziwani ngodya yapakati ya gawolo mu ma radians.
Kukambirana:
Ndizodziwika kuti:
– \( r = 7 \) cm
– Kutalika kwa Arc \( P_b = 11 \text{ cm} \)
Timagwiritsa ntchito njira ya kutalika kwa arc kuti tipeze ngodya \( \theta \):
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
Popeza tapemphedwa kuti tipeze ngodya mu ma radians, timasintha ma radians a 360° ndi ma radians a \(2\pi\):
\[P_b = \theta \times r\]
\[11 = \theta \nthawi 7\]
\[\theta = \frac{11}{7}\]
\[\theta \pafupifupi 1.57 \text{ rad}\]
Kotero, ngodya yapakati ya gawoli ndi ma radians 1.57.
Funso 3:
Bwalo lozungulira la masentimita 16 lili ndi gawo lokhala ndi malo a masentimita 200. Werengani ngodya yapakati ya gawolo.
Kukambirana:
Ndizodziwika kuti:
– \( r = 16 \) cm
– \( L_juring = 200 \text{ cm}^2 \)
Timagwiritsa ntchito fomula ya dera la gawo kuti tipeze \( \theta \):
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[200 = \frac{\theta}{360^\circ} \times \pi \times (16)^2\]
\[200 = \frac{\theta}{360^\circ} \times \pi \times 256\]
\[200 = \frac{\theta \times 256 \times \pi}{360^\circ}\]
\[200 \nthawi 360^\circ = \theta \nthawi 256 \nthawi 3.14\]
\[72000 = \theta \nthawi 256 \nthawi 3.14\]
\[72000 = \theta \nthawi 804.64\]
\[\theta = \frac{72000}{804.64}\]
\[\theta \pafupifupi 89.45^\circ\]
Kotero, ngodya yapakati ya gawoli ndi pafupifupi 89.45°.
Funso 4:
Werengani circumference yonse ya gawo lomwe lili ndi radius ya 12 cm ndi ngodya yapakati ya 120°.
Kukambirana:
Ndizodziwika kuti:
– \( r = 12 \) cm
– \( \theta = 120^\circ \)
Choyamba, tikupeza kutalika kwa arc:
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
\[P_b = \frac{120^\circ}{360^\circ} \nthawi 2\pi \nthawi 12\]
\[P_b = \frac{1}{3} \nthawi 2\pi \nthawi 12\]
\[P_b = 8\pi\malemba{ cm}\]
Kenako, timawerengera circumference ya gawo (kutalika kwa arc + ma radii awiri):
\[K = 2r + P_b\]
\[K = 2 \nthawi 12\malemba{ cm} + 8\pi\malemba{ cm}\]
\[K = 24\malemba{ cm} + 8\pi\malemba{ cm}\]
Ngati \( \pi \) yatengedwa kukhala 3.14, ndiye kuti:
\[K = 24\malemba{cm} + 8 \nthawi 3.14\malemba{cm}\]
\[K = 24\malemba{cm} + 25.12\malemba{cm}\]
\[K = 49.12\malemba{cm}\]
Kotero, circumference yonse ya gawoli ndi 49.12 cm.
Funso 5:
Ngati bwalo lozungulira la 18 cm lili ndi gawo lomwe limapanga ngodya ya 45°, dziwani kutalika kwa arc ndi dera la gawolo.
Kukambirana:
Ndizodziwika kuti:
– \( r = 18 \) cm
– \( \theta = 45^\circ \)
1. Utali wa Arc:
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
\[P_b = \frac{45^\circ}{360^\circ} \times 2\pi \times 18\text{ cm}\]
\[P_b = \frac{1}{8} \times 36\pi\text{ cm}\]
\[P_b = 4.5\pi\malemba{ cm}\]
Ngati \( \pi \) yatengedwa kukhala 3.14, ndiye kuti:
\[P_b = 4.5 \nthawi 3.14\malemba{ cm} \pafupifupi 14.13\malemba{ cm}\]
Kotero, kutalika kwa uta ndi pafupifupi 14.13 cm.
2. Gawo la Gawo:
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[L_juring = \frac{45^\circ}{360^\circ} \times \pi \times (18\text{ cm})^2\]
\[L_juring = \frac{1}{8} \times \pi \times 324\text{ cm}^2\]
\[L_juring = 40.5\pi\text{ cm}^2\]
Ngati \( \pi \) yatengedwa kukhala 3.14, ndiye kuti:
\[L_juring = 40.5 \nthawi 3.14\malemba{ cm}^2 \pafupifupi 127.17\malemba{ cm}^2\]
Kotero, dera la gawoli ndi pafupifupi 127.17 cm².
Mapeto
Munkhaniyi, takambirana zitsanzo zingapo za mavuto okhudza magawo a bwalo ndi mayankho awo. Chofunika kwambiri pakumvetsetsa magawo a bwalo ndi kudziwa bwino njira zoyambira zowerengera dera la gawo ndi kutalika kwa arc. Kuchita mobwerezabwereza komanso kumvetsetsa momwe mungagwiritsire ntchito njirazi pamavuto osiyanasiyana kudzakuthandizani kukonza luso lanu lothana ndi mavuto ofanana.
