Ngā Tauira Pātai me te Kōrero mō ngā Rāngai Porowhita
He kaupapa nui ngā wāhanga o te porowhita i roto i te pāngarau, ā, he maha ngā wā ka puta i roto i ngā whakamātautau me ngā pātai whakaharatau. Ko te wāhanga o te porowhita he rohe e rua ngā radius me te pewa e hono ana i a rāua. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru mō ngā wāhanga o te porowhita, me ngā whakamārama taipitopito, hei whakahōhonu ake i tō mātou māramatanga.
Te Whakamāramatanga o te Wāhanga Porowhita
Ko te wāhanga o te porowhita he wāhanga o te porowhita e herea ana e ngā radius e rua me te pewa kotahi. Ka tatauhia te horahanga o te wāhanga i runga i te haurua o te horahanga katoa o te porowhita. Ko te tātai matua e whakamahia ana hei tatau i tētahi wāhanga koia tēnei:
– Te horahanga o te Juring : \[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
– Te Roa o te Āwhata: \[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
Kei hea:
– Ko te \( \theta \) te rahi o te koki ā-rārangi i roto i ngā nekehanga,
– Ko te radius o te porowhita ko \( r \),
– He pūmau a \( \pi \) (tata ki te 3.14159).
Ngā Pātai Tauira me te Kōrero
Pātai 1:
Homai he porowhita he 10 cm te whānui, me te porowhita he 90° te koki i waenganui. Tātaihia te horahanga o te porowhita.
Kōrero:
E mōhiotia ana:
– \( r = 10 \) henimita
– \( \theta = 90^\circ \)
Ka whakamahia e mātou te tātai mō te horahanga o tētahi rāngai:
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[L_juring = \frac{90^\circ}{360^\circ} \times \pi \times (10\text{ cm})^2\]
\[L_juring = \frac{1}{4} \times \pi \times 100\text{ cm}^2\]
\[Te roa o te hurihanga = 25\pi\kuputuhi{ henimita}^2\]
Mena ka kiia ko te 3.14 te rahi o \( \pi \), kāti:
\[Te roa o te hurihanga = 25 \times 3.14\text{ cm}^2 = 78.5\text{ cm}^2\]
Nō reira, ko te horahanga o te rāngai he 78.5 cm².
Pātai 2:
He 7 cm te whānui o te porowhita, ā, he 11 cm te roa o te pewa. Tāutuhia te koki pokapū o te porowhita i roto i ngā rātiana.
Kōrero:
E mōhiotia ana:
– \( r = 7 \) henimita
– Te roa o te pewa \( P_b = 11 \text{ cm} \)
Ka whakamahia e mātou te tātai roa pewa hei kimi i te koki \( \theta \):
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
Nā te mea e tonoa ana tātou kia kimi i te koki i roto i ngā rātiana, ka whakakapia te 360° ki ngā rātiana \(2\pi\):
\[P_b = \theta \times r\]
\[11 = \theta \ngā 7\]
\[\theta = \frac{11}{7}\]
\[\theta \tata ki te 1.57 \text{ rad}\]
Nō reira, ko te koki pokapū o te rāngai he 1.57 ngā rātiana.
Pātai 3:
He porowhita he 16 cm te whānui, ā, he 200 cm² te horahanga o tētahi porowhita. Tātaihia te koki pokapū o te porowhita.
Kōrero:
E mōhiotia ana:
– \( r = 16 \) henimita
– \( Te roa o te rurunga = 200 \text{ cm}^2 \)
Ka whakamahia e mātou te tātai horahanga mō tētahi rāngai hei kimi i te \( \theta \):
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[200 = \frac{\theta}{360^\circ} \times \pi \times (16)^2\]
\[200 = \frac{\theta}{360^\circ} \times \pi \times 256\]
\[200 = \frac{\theta \times 256 \times \pi}{360^\circ}\]
\[200 \whakareatia ki te 360^\pōro = \theta \whakareatia ki te 256 \whakareatia ki te 3.14\]
\[72000 = \theta \ngā 256 \ngā 3.14\]
\[72000 = \theta \ngā 804.64\]
\[\theta = \frac{72000}{804.64}\]
\[\theta \tata ki te 89.45^\circ\]
Nō reira, ko te koki pokapū o te rāngai he tata ki te 89.45°.
Pātai 4:
Tātaihia te porowhita katoa o tētahi wāhanga he 12 cm te radius, ā, he 120° te koki pokapū.
Kōrero:
E mōhiotia ana:
– \( r = 12 \) henimita
– \( \theta = 120^\circ \)
Tuatahi, ka kitea e tātou te roa o te pewa:
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
\[P_b = \frac{120^\circ}{360^\circ} \times 2\pi \times 12\]
\[P_b = \frac{1}{3} \times 2\pi \times 12\]
\[P_b = 8\pi\kuputuhi{ henimita}\]
Kātahi ka tatauhia te porowhita o te rāngai (roa pewa + rua ngā rātio):
\[K = 2r + P_b\]
\[K = 2 \times 12\kuputuhi{ henimita} + 8\pi\kuputuhi{ henimita}\]
\[K = 24\kuputuhi{ henimita} + 8\pi\kuputuhi{ henimita}\]
Mena ka kiia ko te 3.14 te rahi o \( \pi \), kāti:
\[K = 24\kuputuhi{ henimita} + 8 \times 3.14\kuputuhi{ henimita}\]
\[K = 24\kuputuhi{ henimita} + 25.12\kuputuhi{ henimita}\]
\[K = 49.12\kuputuhi{ henimita}\]
Nō reira, ko te tapeke o te porowhita o te rāngai he 49.12 cm.
Pātai 5:
Mena he 18 cm te whānui o tētahi porowhita, ā, he 45° te koki o te porowhita, tātaihia te roa o te pewa me te horahanga o te porowhita.
Kōrero:
E mōhiotia ana:
– \( r = 18 \) henimita
– \( \theta = 45^\circ \)
1. Te roa o te āwhata:
\[P_b = \frac{\theta}{360^\circ} \times 2\pi r\]
\[P_b = \frac{45^\circ}{360^\circ} \times 2\pi \times 18\text{ cm}\]
\[P_b = \frac{1}{8} \times 36\pi\text{ cm}\]
\[P_b = 4.5\pi\kuputuhi{ henimita}\]
Mena ka kiia ko te 3.14 te rahi o \( \pi \), kāti:
\[P_b = 4.5 \times 3.14\kuputuhi{ henimita} \approx 14.13\kuputuhi{ henimita}\]
Nō reira, ko te roa o te kopere he tata ki te 14.13 henemita.
2. Te Rohe o te Rāngai:
\[L_juring = \frac{\theta}{360^\circ} \times \pi \times r^2\]
\[L_juring = \frac{45^\circ}{360^\circ} \times \pi \times (18\text{ cm})^2\]
\[L_juring = \frac{1}{8} \times \pi \times 324\text{ cm}^2\]
\[Te roa o te hurihanga = 40.5\pi\kuputuhi{ henimita}^2\]
Mena ka kiia ko te 3.14 te rahi o \( \pi \), kāti:
\[Te roa o te hurihanga = 40.5 \times 3.14\text{ cm}^2 \approx 127.17\text{ cm}^2\]
Nō reira, ko te horahanga o te rāngai he tata ki te 127.17 cm².
Whakamutunga
I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira rapanga e pā ana ki ngā wāhanga o te porowhita me ō rātou otinga. Ko te ngako o te mārama ki ngā wāhanga o te porowhita kei roto i te mōhio ki ngā tātai taketake mō te tatau i te horahanga o te wāhanga me te roa o te pewa. Mā te whakaharatau auau me te mārama ki te whakamahi i ēnei tātai ki ngā momo rapanga rerekē, ko te tumanako ka āwhina i a koe ki te whakapai ake i tō pūkenga ki te whakaoti rapanga rite.
