Mienzaniso yeMibvunzo Inokurukura Hukama Huri Pakati Pehurefu HweArc neSector Area
Muzvidzidzo zve geometry, kunyanya mukudzidza madenderedzwa, tinowanzo sangana nepfungwa dzehurefu hwe arc nenzvimbo ye sector. Pfungwa idzi mbiri dzakakosha pakunzwisisa zviitiko zvakasiyana-siyana zve geometrical zvine chekuita nemadenderedzwa. Ngatitangei tatsanangura pfungwa idzi mbiri tisati tapa mienzaniso yezvinetso nemhinduro dzadzo.
Kureba kweArc
Kureba kwearc ndiko kureba kuri pakati pemapoinzi maviri padenderedzwa. Kuti tiverenge kureba kwearc yedenderedzwa, tinowanzoda radius yedenderedzwa (r) ne central angle (θ) iyo arc inonyura nayo muma radians. Fomura yekuverenga kureba kwearc (s) inogona kunyorwa seinotevera:
\[ s = r \times \theta \]
Kana kona yepakati yakapihwa mumadhigirii, tinofanira kutanga taishandura kuita radians ne:
\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \]
Nzvimbo yeSector
Chikamu chikamu chedenderedzwa rakaganhurirwa nema radii maviri uye arc iri pakati pawo. Kuti tiverenge nzvimbo yechikamu, tinoshandisa radius yedenderedzwa (r) ne central angle (θ). Fomura yekuverenga nzvimbo yechikamu (A) ndeiyi:
\[ A = \frac{1}{2} r^2 \times \theta \]
Sezvakaita nehurefu hwe arc, kana kona yepakati ikayerwa nemadhigirii, tinofanira kutanga taishandura kuita radians.
Mibvunzo yemuenzaniso nekukurukurirana
Kuti tijekese pfungwa yehurefu hwearc nenzvimbo yechikamu, ngationgororei mienzaniso yemibvunzo inotevera nehurukuro dzayo.
Mubvunzo 1:
Zvichipiwa denderedzwa rine radius ye 10 cm uye angle yepakati ye 60 degrees, verengai kureba kwe arc nenzvimbo yechikamu chakaumbwa ne angle.
Kukurukurirana:
1. Kuverenga Kureba kweArc:
- Kutanga, tinoshandura kona kubva pamadhigirii kuenda kumaradians:
\[ \theta = 60 \nguva \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
- Kushandisa fomura yekureba kwe arc:
\[ s = r \times \theta \]
\[ s = 10 \times \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Kuverenga Nzvimbo yeSector:
- Kushandisa fomura yenzvimbo yechikamu:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 10^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 100 \times \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Saka, kureba kwe arc i \(\frac{10\pi}{3}\) cm, uye nzvimbo yechikamu i \(\frac{50\pi}{3}\) cm².
Mubvunzo 2:
Denderedzwa rine radius ye7 cm uye kona yepakati yakakomberedzwa ne arc yema radians maviri. Sarudza kureba kwe arc nenzvimbo yechikamu chedenderedzwa.
Kukurukurirana:
1. Kuverenga Kureba kweArc:
- Kona yepakati yatove muma radians, saka tinogona kushandisa zvakananga fomura ye arc length:
\[ s = r \times \theta \]
\[s = 7 \kawa 2 \]
\[ s = 14 \, \chinyorwa{cm} \]
2. Kuverenga Nzvimbo yeSector:
- Kushandisa fomura yenzvimbo yechikamu:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \kawa 7^2 \kawa 2 \]
\[ A = \frac{1}{2} \kawa 49 \kawa 2 \]
\[ A = 49 \, \text{cm}^2 \]
Saka, kureba kwe arc ndeye 14 cm, uye nzvimbo yechikamu ichi ndeye 49 cm².
Mubvunzo 3:
Denderedzwa rine radius ye 12 cm rine sector ine arc length iri 15\(\pi\) cm. Sarudza central angle mumadhigirii nenzvimbo ye sector.
Kukurukurirana:
1. Kuziva Angle Yepakati:
- Kushandisa fomura yehurefu hwe arc kuti uwane kona yepakati:
\[ s = r \times \theta \]
\[ 15\pi = 12 \nguva \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
- Chinja kona yepakati kuita madhigirii:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \text{degrees} \]
2. Kuverenga Nzvimbo yeSector:
- Kushandisa fomura yenzvimbo yechikamu:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 12^2 \times \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \times 144 \times \frac{5\pi}{4} \]
\[ A = 72 \nguva \frac{5\pi}{4} \]
\[ A = 90\pi \, \text{cm}^2 \]
Saka, kona yepakati pechikamu ichi i225 degrees, uye nzvimbo yechikamu ichi i90\(\pi\) cm².
Mhedziso
Kunzwisisa hukama huripo pakati pehurefu hwe arc nenzvimbo yechikamu kunoda kunzwisisa kwakakwana misimboti mikuru yemadenderedzwa uye kushandiswa kwakakodzera kwemafomula. Kuburikidza nematambudziko ekudzidzira ari pamusoro apa, tinogona kuona kukosha kwekudzidzira kushandura angle uye kushandisa zvakananga mafomula mumamiriro e geometry yakatenderera. Danho rega rega muhurukuro yedambudziko rinotibatsira kunzwisisa kuti mafomula anoshanda sei uye kuti tingaashandisa sei zvinobudirira.
Nekuramba tichidzidzira nekunzwisisa zvinhu zvekutanga zvakatsanangurwa, tichava nehunyanzvi mukugadzirisa matambudziko ane chekuita nehurefu hwearc nenzvimbo yechikamu, uye izvi zvichabatsira zvikuru mukushandiswa kwakasiyana-siyana kwemasvomhu nedzimwe nzira dzesainzi.
