Laʻana o kahi Nīnau Kūkākūkā ma nā Piʻo Pōʻai
I ke ʻano honua, he kiʻi mokulele ka pōʻai me nā manaʻo hoihoi he nui e aʻo ai, ʻo kekahi o ia mau mea ʻo ke arc. ʻO ka arc ka ʻāpana o ka lihi o ka pōʻai e waiho ana ma waena o ʻelua mau kiko ma ka pōʻai. Ma kēia ʻatikala, e ʻimi mākou i nā pilikia hoʻohālike like ʻole a me kā lākou mau hoʻonā e pili ana i nā arc.
Ka Hoʻomaopopo Kumu o nā ʻĀkuhi Pōʻai
Ma mua o ka neʻe ʻana i nā nīnau hoʻohālike, he mea nui e hoʻomaopopo mua i kekahi mau manaʻo kumu:
1. Pōʻai:
ʻO ka pōʻai ka hōʻiliʻili ʻana o nā kiko a pau i loko o kahi mokulele e like ka mamao mai kahi kikowaena i hāʻawi ʻia.
2. Radius (Nā Manamana Lima):
ʻO ke radius ka mamao mai ke kikowaena o ka pōʻai a i kekahi kiko ma ka lihi o ka pōʻai.
3. Anawaena:
ʻO ke anawaena ka mamao loa mai kekahi kiko ma ka lihi o kahi pōʻai a i kekahi kiko ma ka ʻaoʻao ʻē aʻe ma o ke kikowaena. ʻO ke anawaena he pālua ia o ka radius.
4. Kūlou:
He ʻāpana o ka lihi o ka pōʻai ka piʻo. Inā aia nā kiko A a me B ma ka lihi o ka pōʻai, a laila ʻo ka piʻo AB ka ʻāpana o ka pōʻai ma waena o A a me B.
Nā Haʻilula e pili ana i nā ʻĀkau o nā Pōʻai
No ke ana ʻana i ka lōʻihi o kahi ʻāʻī, pono mākou e hoʻomaopopo i kekahi mau ʻano hana:
1. Ka Lōʻihi o ke ʻĀkau (L):
ʻO ka lōʻihi o kahi piʻo pōʻai ka lōʻihi o ka lihi o ka pōʻai e komo pū ana me ke piʻo. ʻO ke ʻano hana:
\[
L = r manawa
\]
kahi ʻo \( r \) ka radius o ka pōʻai a ʻo \( \theta \) ke kihi waena ma nā radians e hui pū ana me ke arc.
2. Ka Lōʻihi o ke ʻĀkau ma nā kekelē:
Inā he kekelē ke kihi waena, hoʻohana mākou i ke ʻano hana:
\[
L = \frac{\theta}{360^\circ} \times 2\pi r
\]
kahi ʻo \( \theta \) ke kihi kikowaena ma nā kekelē.
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: Ke helu ʻana i ka lōʻihi o ke arc
Nīnau:
He 10 kenimika ka radius o kahi pōʻai. E helu i ka lōʻihi o ke piʻo i hoʻopaʻa ʻia e kahi kihi kikowaena o 60 degere.
Kūkākūkā:
– Ka laulā o ka pōʻai (\( r \)) = 10 kenimika
– Ke kihi waena (\( \theta \)) = 60°
Ke hoʻohana nei i ke ʻano o ka lōʻihi o ke arc ma nā kekelē:
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
\[
L = \frac{60^\circ}{360^\circ} \times 2 \pi \times 10 \text{ kenimika}
\]
\[
L = \frac{1}{6} \times 2 \pi \times 10 \text{ kenimika}
\]
\[
L = \frac{20\pi}{6} \text{ kenimika}
\]
\[
L \approx 10.47 \text{ kenimika}
\]
No laila, ʻo ka lōʻihi o ke ʻāʻī ma kahi o 10.47 kenimika.
Nīnau 2: Ke hoʻoholo nei i ke kihi waena mai ka lōʻihi o ke aʻo
Nīnau:
Hāʻawi ʻia kahi pōʻai me ka radius o 14 kenimika a he 22 kenimika ka lōʻihi o ke piʻo. E hoʻoholo i ke kihi kikowaena e hui pū ana me ke piʻo ma nā kekelē.
Kūkākūkā:
– Ka laulā o ka pōʻai (\( r \)) = 14 kenimika
– Ka lōʻihi o ke aʻo (\( L \)) = 22 kenimika
E hoʻohana i ke ʻano hana lōʻihi o ke arc e ʻike ai i ka \( \theta \):
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
E pani i nā waiwai i ʻike ʻia:
\[
22 = \frac{\theta}{360^\circ} \times 2 \pi \times 14
\]
Hoʻokaʻawale \( \theta \):
\[
22 = \frac{\theta}{360^\circ} \times 28 \pi
\]
\[
22 manawa 360^\circ = \theta manawa 28 \pi
\]
\[
7920 = \theta \times 28 \pi
\]
\[
\theta = \frac{7920}{28 \pi}
\]
\[
ʻaneʻane 90.72^\circ
\]
No laila, ʻo ka nui o ke kihi waena ma kahi o 90.72 degere.
Nīnau 3: Ke helu ʻana i ka ʻāpana o kahi ʻāpana
Nīnau:
Ua hoʻokumu ʻia kahi ʻāpana o ka pōʻai e kahi kihi kikowaena o 120 degere me ka radius o 7 cm. E hoʻoholo i ka ʻāpana o ka ʻāpana.
Kūkākūkā:
– Ka laulā o ka pōʻai (\( r \)) = 7 kenimika
– Ke kihi waena (\( \theta \)) = 120°
E hoʻohana i ke ʻano no ka wahi o kahi ʻāpana:
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
E pani i nā waiwai i ʻike ʻia:
\[
A = \frac{120^\circ}{360^\circ} \times \pi \times 7^2
\]
\[
A = \frac{1}{3} \times \pi \times 49
\]
\[
A = \frac{49\pi}{3}
\]
\[
He \approx 51.43 \text{ cm}^2
\]
No laila, ʻo ka nui o ka ʻāpana ma kahi o 51.43 cm².
Nīnau 4: Ke hoʻoholo nei i ka Arc mai ka ʻĀpana o kahi ʻĀpana
Nīnau:
ʻO ka ʻāpana o kahi ʻāpana o ka pōʻai me ka radius o 6 cm he 18π cm². He aha ka lōʻihi o ke piʻo o ka ʻāpana?
Kūkākūkā:
– Ka laulā o ka pōʻai (\( r \)) = 6 kenimika
– ʻĀpana o ka ʻāpana (\( A \)) = 18π cm²
E hoʻohana i ke ʻano hana no kahi ʻāpana e ʻike ai i ke kihi waena \( \theta \):
\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]
E pani i nā waiwai i ʻike ʻia:
\[
18\pi = \frac{\theta}{360^\circ} \times \pi \times 6^2
\]
\[
18\pi = \frac{\theta}{360^\circ} \times 36\pi
\]
\[
18 = \frac{\theta}{360^\circ} \times 36
\]
\[
18 manawa 360^\circ = \theta manawa 36
\]
\[
6480 = \theta \times 36
\]
\[
\theta = \frac{6480}{36}
\]
\[
\theta = 180^\circ
\]
I kēia manawa, me kahi kihi kikowaena o 180 degere, hoʻoholo mākou i ka lōʻihi o ke arc:
\[
L = \frac{\theta}{360^\circ} \times 2 \pi r
\]
\[
L = \frac{180^\circ}{360^\circ} \times 2 \pi \times 6
\]
\[
L = \frac{1}{2} \times 2 \pi \times 6
\]
\[
L = \pi \times 6
\]
\[
L \approx 18.85 \text{ kenimika}
\]
No laila, ʻo ka lōʻihi o ke kakaka ma kahi o 18.85 kenimika.
Ka hopena
ʻO ka hoʻomaopopo ʻana i nā piʻo pōʻai a pehea e helu ai iā lākou he kumu koʻikoʻi ia i ke geometry a me ka makemakika ma ke ʻano laulā. Ma o nā laʻana i kūkākūkā ʻia ma kēia ʻatikala, manaʻo ʻia e loaʻa i ka poʻe heluhelu kahi ʻike maikaʻi aʻe i ke ʻano o ka helu ʻana i ka lōʻihi o ke piʻo, ka ʻāpana ʻāpana, a me ka hoʻoholo ʻana i ke kihi waena e hui pū ana me kahi piʻo. ʻO ka hoʻomaopopo maikaʻi ʻana i kēia mau manaʻo kumu e kōkua nui i ka hoʻoponopono ʻana i nā pilikia like ʻole e pili ana i nā pōʻai.
