Zitsanzo za Mafunso Okhudza Ubale Pakati pa Kutalika kwa Arc ndi Gawo la Gawo
Mu maphunziro a geometry, makamaka pophunzira mabwalo, nthawi zambiri timakumana ndi malingaliro a kutalika kwa arc ndi dera la gawo. Malingaliro awiriwa ndi ofunikira kwambiri pomvetsetsa zochitika zosiyanasiyana za geometry zokhudzana ndi mabwalo. Choyamba tiyeni tifotokoze mfundo ziwirizi tisanapereke zitsanzo za mavuto ndi mayankho awo.
Utali wa Arc
Kutalika kwa arc ndi mtunda womwe uli pakati pa mfundo ziwiri pa bwalo. Kuti tiwerenge kutalika kwa arc ya bwalo, nthawi zambiri timafunikira radius ya bwalo (r) ndi ngodya yapakati (θ) yomwe arc imadutsa mu ma radians. Njira yowerengera kutalika kwa arc (s) ikhoza kulembedwa motere:
\[ s = r \times \theta \]
Ngati ngodya yapakati yaperekedwa mu madigiri, choyamba tiyenera kuisintha kukhala ma radians ndi:
\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \]
Chigawo cha Gawo
Gawo ndi gawo la bwalo lozunguliridwa ndi ma radii awiri ndi arc pakati pawo. Kuti tiwerengere dera la gawo, timagwiritsa ntchito radius ya bwalo (r) ndi ngodya yapakati (θ). Fomula yowerengera dera la gawo (A) ndi iyi:
\[ A = \frac{1}{2} r^2 \times \theta \]
Monga momwe zilili ndi kutalika kwa arc, ngati ngodya yapakati imayesedwa mu madigiri, choyamba tiyenera kuisintha kukhala ma radians.
Mafunso ndi Kukambirana Zitsanzo
Kuti timvetse bwino tanthauzo la kutalika kwa arc ndi gawo la gawo, tiyeni tiwone zitsanzo za mafunso awa ndi zokambirana zawo.
Funso 1:
Popeza muli ndi bwalo lozungulira la 10 cm ndi ngodya yapakati ya madigiri 60, werengani kutalika kwa arc ndi dera la gawo lomwe limapangidwira ndi ngodyayo.
Kukambirana:
1. Kuwerengera Utali wa Arc:
Choyamba, timasintha ngodya kuchokera ku madigiri kupita ku ma radians:
\[ \theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
- Pogwiritsa ntchito njira yowerengera kutalika kwa arc:
\[ s = r \times \theta \]
\[ s = 10 \times \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Kuwerengera Dera la Gawo:
- Kugwiritsa ntchito njira ya gawo la gawo:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 10^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 100 \times \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Kotero, kutalika kwa arc ndi \(\frac{10\pi}{3}\) cm, ndipo dera la gawoli ndi \(\frac{50\pi}{3}\) cm².
Funso 2:
Bwalo lili ndi utali wa masentimita 7 ndi ngodya yapakati yomwe imayendetsedwa ndi utali wa ma radians awiri. Dziwani kutalika kwa utali ndi dera la gawo la bwalo.
Kukambirana:
1. Kuwerengera Utali wa Arc:
– Ngodya yapakati ili kale mu ma radians, kotero titha kugwiritsa ntchito mwachindunji njira ya kutalika kwa arc:
\[ s = r \times \theta \]
\[s = 7 \nthawi 2 \]
\[ s = 14 \, \malemba{cm} \]
2. Kuwerengera Dera la Gawo:
- Kugwiritsa ntchito njira ya gawo la gawo:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \nthawi 7^2 \nthawi 2 \]
\[ A = \frac{1}{2} \nthawi 49 \nthawi 2 \]
\[ A = 49 \, \malemba{cm}^2 \]
Choncho, kutalika kwa arc ndi 14 cm, ndipo dera la gawoli ndi 49 cm².
Funso 3:
Bwalo lozungulira la 12 cm lili ndi gawo lomwe kutalika kwake kwa arc ndi 15\(\pi\) cm. Dziwani ngodya yapakati mu madigiri ndi dera la gawolo.
Kukambirana:
1. Kudziwa Ngodya Yapakati:
- Kugwiritsa ntchito njira yowerengera kutalika kwa arc kuti mupeze ngodya yapakati:
\[ s = r \times \theta \]
\[ 15\pi = 12 \nthawi \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
- Sinthani ngodya yapakati kukhala madigiri:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \malemba{madigiri} \]
2. Kuwerengera Dera la Gawo:
- Kugwiritsa ntchito njira ya gawo la gawo:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 12^2 \times \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \times 144 \times \frac{5\pi}{4} \]
\[ A = 72 \times \frac{5\pi}{4} \]
\[ A = 90\pi \, \malemba{cm}^2 \]
Kotero, ngodya yapakati ya gawoli ndi madigiri 225, ndipo dera la gawoli ndi 90\(\pi\) cm².
Mapeto
Kumvetsetsa ubale womwe ulipo pakati pa kutalika kwa arc ndi gawo la gawo kumafuna kumvetsetsa bwino mfundo zazikulu za mabwalo ndi kugwiritsa ntchito bwino ma formula. Kudzera mu mavuto omwe ali pamwambapa, titha kuwona kufunika kodziwa bwino kusintha kwa ngodya ndikugwiritsa ntchito ma formula mwachindunji pankhani ya geometry yozungulira. Gawo lililonse mu kukambirana za vuto limatithandiza kumvetsetsa momwe ma formula amagwirira ntchito komanso momwe tingawagwiritsire ntchito bwino.
Mwa kupitiriza kuchita ndikumvetsetsa zoyambira zomwe zafotokozedwa, tidzakhala aluso kwambiri pakuthana ndi mavuto okhudzana ndi kutalika kwa arc ndi gawo, ndipo izi zidzakhala zothandiza kwambiri pamasamu osiyanasiyana ndi ntchito zina zasayansi.
