Ngā Tauira Pātai e Matapaki ana i ngā Tauwehenga Pāngarau i roto i ngā Piramiti
Pendahuluan
Ko te tikanga, ko te ine ine he peka o te pāngarau e ako ana i ngā whanaungatanga i waenga i ngā koki me ngā taha o ngā tapatoru. He mea nui ngā ōwehenga ine ine i roto i ngā momo mara pēnei i te ahupūngao, te hangarau, tae noa ki te hoahoanga. Ko tētahi o ngā tauira pai rawa atu o te whakamahinga o te ine ine i roto i te hoahoanga ko ngā Piramiti o Īhipa. I roto i tēnei tuhinga, ka matapakihia e mātou ngā ōwehenga ine ine ine mā te whakamahi i ngā tauira e pā ana ki ngā piramiti.
He Kupu Whakataki ki te Ine Taurite i roto i ngā Piramiti
He tino rongonui ngā piramita o Īhipa, inā koa ngā piramita o Kiza, ā, kua rangahaua e te tini o ngā tohunga pāngarau me ngā kaihoahoa whare. Ko tētahi o ngā wāhanga nui o te piramita ko te tapatoru. Ka kitea ngā tapatoru i roto i ngā tirohanga kōtaha me ngā wāhanga whakawhiti.
Mai i tētahi pīramiti, ka kitea e tātou ngā tapatoru hāngai, ngā tapatoru ōrite, me ētahi atu āhua tapatoru. He tino āwhina te whakamahinga o te ine whārite ki te whakatau i ngā ine, te teitei, me te pari o tētahi pīramiti.
Tauira raruraru
Pātai 1: Te Tatau i te Teitei o te Piramiti
"Me kī he 150 mita te roa o te turanga o tētahi piramiti, ā, he 130 mita te taha piko (apothem). He aha te teitei o te piramiti?"
Kōrero:
I tēnei raruraru, ka hoatu ki a tātou te hypotenuse me te roa o te turanga o te piramiti. Hei tatau i te teitei o te piramiti, ka taea e tātou te whakamahi i te ariā Pythagorean. Ka taea te wehewehe i te piramiti kia rua ngā tapatoru matau.
1. Me kimi e tātou te haurua o te roa o te taha turanga hei hanga i tētahi tapatoru hāngai.
\( \text{Te haurua o te roa o te taha turanga} = \frac{150}{2} = 75 \text{ mita} \)
2. E mōhio ana mātou:
\( a^2 + b^2 = c^2 \)
ko \(a\) te haurua o te roa o te taha turanga, ko \(b\) te teitei o te piramiti, ā, ko \(c\) te hypotenuse.
3. Tāpirihia ngā tau ki te whārite:
\( 75^2 + b^2 = 130^2 \)
4. Tātaihia:
\( 5625 + b^2 = 16900 \)
\( b^2 = 16900 – 5625 \)
\( b^2 = 11275 \)
\( b = \sqrt{11275} \approx 106.2 \text{ mita} \)
Nō reira, ko te teitei o te piramiti he tata ki te 106.2 mita.
Pātai 2: Te Tatau i te Koki o te Piramiti
"He aha te koki o te piko o te apothema o te piramiti ki te turanga o te piramiti he 150 mita te roa o te taha turanga, ā, 106.2 mita te teitei?"
Kōrero:
Hei kimi i te koki o te piko (\(\theta\)) o te apothema ki te turanga o te piramiti, ka taea e tātou te whakamahi i te mahi pātōtō, arā, te pāngent (\(\tan\)).
1. Whakamahia te tātai \(\tan(\theta) = \frac{\text{height}}{\frac{\text{base}}{2}}\).
2. Tāuruhia ngā tau:
\( \tan(\theta) = \frac{106.2}{75} \)
3. Tātaihia:
\( \tan(\theta) \tata ki te 1.416 \)
4. Kimihia te koki mā te whakamahi i te tānga whakamuri (\(\tan^{-1}\)):
\( \theta = \tan^{-1}(1.416) \tata ki te 54.14^\circ \)
Nō reira, ko te koki o te piko o te apothem ki te turanga o te piramiti he 54.14 nekehanga pea.
Pātai 3: Te Tātai i te Roa o te Apothem mā te Sine me te Cosine
"Me kī he 120 mita te teitei o tētahi piramiti, ā, ko te koki o te piko o te apothema o te piramiti ki te turanga he 55 nekehanga. He aha te roa o te apothema?"
Kōrero:
Ka taea e tātou te whakamahi i te mahi sine, te mahi cosine rānei hei whakaoti i tēnei raruraru.
1. Whakamahia te kōsina hei whakaoti, me te maumahara:
\( \cos(\theta) = \frac{\text{Patata}}{\text{Hypotenuse}} \)
2. Whakaritea anō te whārite mō te Hypotenuse (apothem):
\( \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \)
3. Tāuruhia ngā tau:
\( \text{Hypotenuse} = \frac{120}{\cos(55^\circ)} \)
4. Tātaihia:
\( \cos(55^\circ) \tata ki te 0.5736 \)
\( \text{Hypotenuse} = \frac{120}{0.5736} \approx 209.3 \text{ mita} \)
Nō reira, ko te roa o te apothema he tata ki te 209.3 mita.
Whakamutunga
I roto i ngā raruraru i runga ake nei, kua whakamahia e mātou ngā ōwehenga pākoki hei tatau i te teitei, te koki pari, me te roa āpōtema o tētahi pīramiti. Mā te mārama ki te pākoki, ka taea e tātou te whakaoti rapanga āhuahanga maha e ahua uaua ana i te tirohanga tuatahi. He taputapu tino nui te pākoki hei mārama me te whakaoti rapanga e tūtakihia ana e tātou i te ao tūturu, inā koa i roto i ngā horopaki hoahoanga pērā i ngā pīramiti o Īhipa.
