Tauira Pātai me te Kōrero mō ngā Mahi Taupūnga
He ariā taketake ngā mahi taupū i roto i te pāngarau, e kapi ana i te huringa taupū, te tipu taupū me te pirau taupū. He maha ngā whakamahinga mahi a te māramatanga hōhonu ki ēnei mahi i te ao tūturu, mai i te matū me te ahupūngao ki te koiora me te ōhanga. Ka tūhuratia e tēnei tuhinga ētahi tauira o ngā mahi taupū me ā rātou otinga hei āwhina i a koe ki te mārama ake ki tēnei kaupapa.
Whakataki ki ngā Mahi Taupū
Ko te āhua whānui o tētahi mahi taupūnga ko \( y = a \cdot b^x \), arā:
– Ko te uara mahi ko \( y \)
– He pūmau te \( a \)
– Ko te pūtake taupūnga \( b \)
– Ko te taurangi motuhake ko \( x \)
I te nuinga o te wā, ki te mea ko te \( b > 1 \), ka pāngia te mahi e te tipu taupū, ā, ki te mea ko te \( 0 < b < 1 \), ka pāngia te mahi e te pirau taupū. Tauira Raru me ngā Mahi Taupū Anei ētahi tauira raru hei whakaatu i te whakamahinga o ngā mahi taupū me tā rātou matapakinga taipitopito. Tauira Raru 1: Raru Tipu Taupori: He 500 ngā rauropi o tētahi taupori kitakita, ā, kei te whakarea i te tere ka taea te whakatauira e te mahi taupū \( P(t) = 500 \cdot 2^t \), ina inehia te \( t \) i roto i ngā hāora. He aha te taupori kitakita i muri i te 5 hāora?
Kōrero: I roto i tēnei raruraru, e mōhio ana tātou: - Taupori tīmatanga, \( P_0 = 500 \) - \( b = 2 \) - \( t = 5 \) Me whakamahi noa i te uara o \( t \) ki te mahi taupū kua hoatu: \[ P(5) = 500 \cdot 2^5 \] Te Tātai \( 2^5 \): \[ 2^5 = 32 \] Inaianei, whakareatia ki te taupori tīmatanga: \[ P(5) = 500 \cdot 32 = 16000 \] Nō reira, ko te taupori kitakita i muri i te 5 hāora he 16.000 ngā rauropi. Tauira Raru 2: Raru Pirau Irahiko: He 200 karamu o tētahi matū kei roto i tētahi tauira irahiko me te haurua-ora o te 3 hāora. Ko te mahi taupū e whakaahua ana i te nui o te matū e toe ana i muri i ngā haora \( t \) ko \( N(t) = 200 \cdot \left(\frac{1}{2}\right)^{t/3} \). E hia te matū e toe ana i muri i ngā haora 9? Otinga: I roto i tēnei raruraru, e mōhio ana tātou: - Papatipu tīmatanga, \( N_0 = 200 \) karamu - Pūtake o te taupū, \( b = \frac{1}{2} \) - \( t = 9 \) Ka whakakapia e tātou te uara o \( t = 9 \) ki roto i te mahi taupū: \[ N(9) = 200 \cdot \left(\frac{1}{2}\right)^{9/3} \] Whakangāwaritia ngā taupū: \[ 9/3 = 3 \] Nō reira ka pēnei te mahi: \[ N(9) = 200 \cdot \left(\frac{1}{2}\right)^3 \] Te Tātai \( \left( \frac{1}{2} \right)^3 \): \[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \] Nā, whakareatia ki te papatipu tīmatanga: \[ N(9) = 200 \cdot \frac{1}{8} = 25 \] Nō reira, ko te nui o te matū e toe ana i muri i te 9 hāora he 25 karamu. Tauira Raru 3: Raru Tipu Ōhanga: Ka pāngia tētahi whenua e te tipu ōhanga o te 4% ia tau, ka taea te whakatauira mā te mahi taupūnga \( G(t) = G_0 \cdot (1.04)^t \), ko \( G_0 \) te GDP tuatahi, ā, ko \( t \) te wā i roto i ngā tau. Mena ko te GDP tuatahi ko \( G_0 = 1.000.000 \), he aha te GDP i muri i ngā tau e 7? Otinga: Kua hoatu: - GDP tīmatanga, \( G_0 = 1.000.000 \) - Tere tipu, \( b = 1.04 \) - \( t = 7 \) Ka whakakapia e mātou te uara o \( t = 7 \) ki roto i te mahi taupū: \[ G(7) = 1.000.000 \cdot (1.04)^7 \] Te tatau \( (1.04)^7 \): \[ (1.04)^7 \approx 1.316074 \] Inaianei, whakareatia ki te GDP tīmatanga: \[ G(7) = 1.000.000 \cdot 1.316074 \approx 1.316.074 \] Nō reira, ko te GDP i muri i ngā tau e 7 e kiia ana he tata ki te 1.316.074. Tauira Raru 4: Raru Uara Pūtea: Ka taea te whakatauira i tētahi haumitanga tuatahi o te 20.000 me te reiti huamoni ā-tau o te 5% e whakamahia ana me te whakakotahitanga ā-tau e te mahi \( A(t) = 20000 \cdot (1+0.05)^t \), ko \( A(t) \) te uara katoa o te haumitanga i muri i ngā tau \( t \). Tātaihia te uara o te haumitanga i muri i ngā tau 10. Otinga: Hoatu: - Te haumitanga tuatahi, \( A_0 = 20000 \) - Te reiti huamoni ā-tau, \( b = 1.05 \) - \( t = 10 \) Ka whakakapia e mātou te uara o \( t = 10 \) ki roto i te mahi taupū: \[ A(10) = 20000 \cdot (1.05)^{10} \] Te tatau \( (1.05)^{10} \): \[ (1.05)^{10} \approx 1.62889 \] Inaianei, whakareatia ki te haumitanga tuatahi: \[ A(10) = 20000 \cdot 1.62889 \approx 32.577,80 \] Nō reira, ko te uara o te haumitanga i muri i te 10 tau he tata ki te 32.577,80. He taputapu kaha ngā mahi taupū i roto i te pāngarau me te whānuitanga o ngā tono mahi. Mai i te tipu o te taupori ki te pirau irahiko me te tipu ōhanga, he mea nui te mārama me te whakamahi i ngā mahi taupū. Mā te matapaki i ngā tauira pēnei i te mea i runga ake nei ka āwhina i te whakamārama i ngā ariā me te whakapai ake i ngā pūkenga whakaoti rapanga. Me haere tonu te whakaharatau me te tūhura i ngā whakamahinga o ngā mahi taupū hei hōhonu ake i tō māramatanga.
