Imibuzo Eyisibonelo Exoxa Ngobudlelwano Phakathi Kobude Be-Arc kanye Nendawo Yomkhakha
Ezifundweni zejiyometri, ikakhulukazi ekufundweni kwezindilinga, sivame ukuhlangana nemibono yobude be-arc nendawo yesigaba. Le mibono emibili ibalulekile ekuqondeni izimo ezahlukahlukene zejiyometri ezihilela izindilinga. Ake siqale sichaze le mibono emibili ngaphambi kokunikeza izibonelo zezinkinga kanye nezixazululo zazo.
Ubude be-Arc
Ubude be-arc yibanga elihambisana ne-arc phakathi kwamaphuzu amabili endilinganeni. Ukuze sibale ubude be-arc yendilinga, sivame ukudinga i-radius yendilinga (r) kanye ne-engeli ephakathi (θ) lapho i-arc idlula khona ngama-radians. Ifomula yokubala ubude be-arc (s) ingabhalwa kanje:
\[ s = r \times \theta \]
Uma i-engeli ephakathi inikezwa ngamadigri, kumele siqale siyiguqulele kuma-radians ngokuthi:
\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \]
Indawo Yomkhakha
Isigaba siyingxenye yendilinga eboshwe ngama-radii amabili kanye ne-arc ephakathi kwawo. Ukuze sibale indawo yesigaba, sisebenzisa i-radius yendilinga (r) kanye ne-engeli ephakathi (θ). Ifomula yokubala indawo yesigaba (A) yile:
\[ A = \frac{1}{2} r^2 \times \theta \]
Njengoba kunjalo ngobude be-arc, uma i-engeli ephakathi ilinganiswa ngamadigri, kumelwe siqale siyiguqule ibe ama-radian.
Imibuzo Eyisibonelo Nengxoxo
Ukuze sicacise umqondo wobude be-arc kanye nendawo yesigaba, ake sibukeze imibuzo elandelayo eyisibonelo kanye nezingxoxo zayo.
Umbuzo 1:
Uma unikezwe indilinga enobubanzi obungu-10 cm kanye ne-engeli ephakathi engama-degrees angu-60, bala ubude be-arc kanye nendawo yomkhakha eyakhiwe yi-engeli.
Ingxoxo:
1. Ukubala Ubude be-Arc:
– Okokuqala, siguqula i-engeli kusukela kumadigri ibe ama-radian:
\[ \theta = 60 \izikhathi \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
- Ukusebenzisa ifomula yobude be-arc:
\[ s = r \times \theta \]
\[ s = 10 \izikhathi \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Ukubala Indawo Yomkhakha:
– Ukusebenzisa ifomula yendawo yomkhakha:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 10^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 100 \times \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Ngakho-ke, ubude be-arc buyi-\(\frac{10\pi}{3}\) cm, kanti indawo yomkhakha ingu-\(\frac{50\pi}{3}\) cm².
Umbuzo 2:
Indilinga inobubanzi obungu-7 cm kanye ne-engeli ephakathi engaphansi kwe-arc yama-radian amabili. Nquma ubude be-arc kanye nendawo yomkhakha wendilinga.
Ingxoxo:
1. Ukubala Ubude be-Arc:
– I-engeli ephakathi isivele ingama-radian, ngakho-ke singasebenzisa ngqo ifomula yobude be-arc:
\[ s = r \times \theta \]
\[s = 7 \izikhathi 2 \]
\[ s = 14 \, \umbhalo{cm} \]
2. Ukubala Indawo Yomkhakha:
– Ukusebenzisa ifomula yendawo yomkhakha:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \izikhathi 7^2 \izikhathi 2 \]
\[ A = \frac{1}{2} \izikhathi ezingu-49 \izikhathi ezingu-2 \]
\[ A = 49 \, \umbhalo{cm}^2 \]
Ngakho-ke, ubude be-arc buyi-14 cm, kanti indawo yomkhakha ingu-49 cm².
Umbuzo 3:
Isiyingi esine-radius engu-12 cm sinomkhakha obude bawo obungu-15\(\pi\) cm. Thola i-engeli ephakathi ngamadigri nendawo yomkhakha.
Ingxoxo:
1. Ukunquma i-Engela Ephakathi:
– Ukusebenzisa ifomula yobude be-arc ukuthola i-engeli ephakathi:
\[ s = r \times \theta \]
\[ 15\pi = 12 \izikhathi \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
- Guqula i-engeli ephakathi ibe amadigri:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \text{degrees} \]
2. Ukubala Indawo Yomkhakha:
– Ukusebenzisa ifomula yendawo yomkhakha:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 12^2 \times \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \times 144 \times \frac{5\pi}{4} \]
\[ A = 72 \izikhathi \frac{5\pi}{4} \]
\[ A = 90\pi \, \text{cm}^2 \]
Ngakho-ke, i-engeli ephakathi yomkhakha ingama-degree angu-225, kanti indawo yomkhakha ingama-90\(\pi\) cm².
Isiphetho
Ukuqonda ubudlelwano phakathi kobude be-arc nendawo yesigaba kudinga ukuqonda okujulile izimiso eziyisisekelo zezindilinga kanye nokusetshenziswa okufanele kwamafomula. Ngezinkinga zokuzijwayeza ezingenhla, singabona ukubaluleka kokuqonda ukuguqulwa kwama-engeli nokusebenzisa ngqo amafomula kumongo we-geometry eyindilinga. Isinyathelo ngasinye engxoxweni yenkinga sisisiza ukuqonda ukuthi amafomula asebenza kanjani nokuthi singawasebenzisa kanjani ngempumelelo.
Ngokuqhubeka nokuzijwayeza nokuqonda izisekelo ezichazwe, sizoba nekhono lokuxazulula izinkinga ezihilela ubude be-arc nendawo yomkhakha, futhi lokhu kuzoba usizo kakhulu ezinhlotsheni ezahlukene zezibalo nezinye izinhlelo zokusebenza zesayensi.
