Ngā Tauira Pātai e Matapaki ana i te Hononga i waenganui i te Roa o te Āwhata me te Horahanga o te Rārangi
I roto i ngā akoranga āhuahanga, inā koa i roto i te ako i ngā porowhita, he maha ngā wā ka tūtaki tātou ki ngā ariā o te roa o te pewa me te horahanga ā-wāhanga. He mea nui ēnei ariā e rua hei mārama ki ngā āhuatanga āhuahanga e pā ana ki ngā porowhita. Me whakamārama tuatahi tātou i ēnei ariā e rua i mua i te hoatu i ngā tauira rapanga me ō rātou otinga.
Te roa o te pewa
Ko te roa o te pewa ko te tawhiti i te taha o te pewa i waenganui i ngā pūwāhi e rua o te porowhita. Hei tatau i te roa o te pewa o te porowhita, me whakamahi te radius o te porowhita (r) me te koki pokapū (θ) e anga atu ai te pewa ki roto i ngā rātiana. Ka taea te tuhi i te tātai mō te tatau i te roa o te pewa (s) penei:
\[ s = r \times \theta \]
Mena ka hoatuhia te koki pokapū i roto i ngā nekehanga, me huri tuatahi tātou ki ngā rātiana mā te:
\[ \theta_{radians} = \theta_{nekehanga} \times \frac{\pi}{180} \]
Rohe o te Rāngai
Ko te rāngai he wāhanga o tētahi porowhita e herea ana e ngā radius e rua me te pewa i waenganui i a rāua. Hei tatau i te horahanga o tētahi rāngai, ka whakamahia e mātou te radius o te porowhita (r) me te koki pokapū (θ). Ko te tātai mō te tatau i te horahanga o tētahi rāngai (A) ko:
\[ A = \frac{1}{2} r^2 \times \theta \]
Pērā i te roa o te pewa, ki te inehia te koki pokapū i roto i ngā nekehanga, me huri tuatahi ki ngā rātiana.
Ngā Pātai Tauira me te Kōrero
Hei whakamārama i te ariā o te roa o te pewa me te horahanga o te rāngai, me arotake ngā tauira pātai e whai ake nei me ā rātou matapakinga.
Pātai 1:
Homai he porowhita he 10 cm te whānui, ā, he 60 nekehanga te koki i waenganui, tatauhia te roa o te pewa me te horahanga o te rāngai i hangaia e te koki.
Kōrero:
1. Te Tātai i te Roa o te Āwhata:
– Tuatahi, ka hurihia e tātou te koki mai i ngā nekehanga ki ngā rātiana:
\[ \theta = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \, \text{radian} \]
– Mā te whakamahi i te tātai roa pewa:
\[ s = r \times \theta \]
\[ s = 10 \times \frac{\pi}{3} \]
\[ s = \frac{10\pi}{3} \, \text{cm} \]
2. Te Tatau i te Horahanga o tētahi Wāhanga:
– Mā te whakamahi i te tātai mō te horahanga o tētahi rāngai:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 10^2 \times \frac{\pi}{3} \]
\[ A = \frac{1}{2} \times 100 \times \frac{\pi}{3} \]
\[ A = \frac{100\pi}{6} \]
\[ A = \frac{50\pi}{3} \, \text{cm}^2 \]
Nō reira, ko te roa o te pewa ko \(\frac{10\pi}{3}\) cm, ā, ko te horahanga o te rāngai ko \(\frac{50\pi}{3}\) cm².
Pātai 2:
He 7 cm te whānui o te porowhita, ā, e 2 ngā rātiana te whānui o te koki pokapū e tautokona ana e te pewa. Tātaihia te roa o te pewa me te horahanga o te wāhanga o te porowhita.
Kōrero:
1. Te Tātai i te Roa o te Āwhata:
– Kei roto kē te koki pokapū i ngā rātiana, nō reira ka taea e tātou te whakamahi tika i te tātai roa pewa:
\[ s = r \times \theta \]
\[ s = 7 \whakareatia ki te 2 \]
\[ s = 14 \, \kuputuhi{cm} \]
2. Te Tatau i te Horahanga o tētahi Wāhanga:
– Mā te whakamahi i te tātai mō te horahanga o tētahi rāngai:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 7^2 \times 2 \]
\[ A = \frac{1}{2} \times 49 \times 2 \]
\[ A = 49 \, \text{cm}^2 \]
Nō reira, ko te roa o te pewa he 14 cm, ā, ko te horahanga o te rāngai he 49 cm².
Pātai 3:
He porowhita he 12 cm te whānui, ā, ko te roa o te pewa he 15\(\pi\) cm. Tātaihia te koki pokapū i roto i ngā nekehanga me te horahanga o te porowhita.
Kōrero:
1. Te whakatau i te koki pokapū:
– Mā te whakamahi i te tātai roa pewa hei kimi i te koki pokapū:
\[ s = r \times \theta \]
\[ 15\pi = 12 \times \theta \]
\[ \theta = \frac{15\pi}{12} \]
\[ \theta = \frac{5\pi}{4} \, \text{radian} \]
– Tahurihia te koki pokapū ki ngā nekehanga:
\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} \]
\[ \theta = \frac{5 \times 180}{4} \]
\[ \theta = 225 \, \text{ngā nekehanga} \]
2. Te Tatau i te Horahanga o tētahi Wāhanga:
– Mā te whakamahi i te tātai mō te horahanga o tētahi rāngai:
\[ A = \frac{1}{2} r^2 \times \theta \]
\[ A = \frac{1}{2} \times 12^2 \times \frac{5\pi}{4} \]
\[ A = \frac{1}{2} \times 144 \times \frac{5\pi}{4} \]
\[ A = 72 \times \frac{5\pi}{4} \]
\[ A = 90\pi \, \text{cm}^2 \]
Nō reira, ko te koki pokapū o te rāngai he 225 nekehanga, ā, ko te horahanga o te rāngai he 90\(\pi\) cm².
Whakamutunga
Hei mārama ki te whanaungatanga i waenga i te roa o te pewa me te horahanga o te wāhanga, me mārama pai ki ngā mātāpono taketake o ngā porowhita me te whakamahinga tika o ngā tātai. Mā roto i ngā mahi whakaharatau i runga ake nei, ka kitea te hiranga o te matatau ki ngā huringa koki me te whakamahi tika i ngā tātai i roto i te horopaki o te āhuahanga porowhita. Mā ia taahiraa o te matapakinga o te raruraru ka āwhina i a tātou ki te mārama ki te mahi a ngā tātai me te whakamahi pai i aua tātai.
Mā te whakaharatau tonu me te mārama ki ngā kaupapa matua kua whakamāramahia, ka matatau ake tātou ki te whakaoti rapanga e pā ana ki te roa o te pewa me te horahanga o te rāngai, ā, ka tino whai hua tēnei i roto i ngā momo whakamahinga pāngarau me ngā mahi pūtaiao kē atu.
