Ngā Tauira Pātai e Matapaki ana i te Tipu Taupūtanga
He ariā matua te tipu taupū i roto i ngā mara maha, tae atu ki te ōhanga, te koiora, te pakihi, me te ahupūngao. I roto i te oranga o ia rā, he maha ngā wā ka tūtaki tātou ki ngā āhuatanga e pā ana ki te tipu taupū, pērā i te tipu o te taupori tangata, te hekenga o ngā uara ingoa nā te pikinga o te utu, te horapa o ngā huaketo, me te maha atu. Ka matapakihia e tēnei tuhinga ngā tauira raruraru me ngā otinga e pā ana ki te tipu taupū, hei āwhina i ngā kaipānui kia mārama ake ki tēnei ariā.
Ngā Ariā Taketake o te Tipu Taupūtanga
Ko te tipu taupū e tautuhia ana ko te tipu e puta ana i te tere pumau mō ia waeine wā. Mā te pāngarau, ka taea te tautuhi i te tipu taupū mā te whārite:
\[ N(t) = N_0 \cdot e^{kt} \]
Kei hea:
– Ko te tapeke o te \( N(t) \) i te wā \( t \)
– Ko te tau tīmatanga ko \( N_0 \)
– Ko te tere tipu te \( k \)
– ko te wā ko te \( t \)
– Ko te tau a Euler te \( e \) (tata ki te 2.718)
Mena he pai te uara o \( k \), ka puta he tipu taupū. I tetahi atu taha, ki te mea he kino te uara o \( k \), ka puta he whakaheke taupū.
Tauira Pātai 1: Te Tipu o te Taupori
Pātai
Ko te taupori o tētahi tāone nui he 100,000 i te tīmatanga. Mēnā ka piki te taupori o te tāone nui i te tere tere o te 2% ia tau, ka hia te taupori o te tāone nui i muri i ngā tau 10?
Te Whakatau
Tuatahi, me tautuhi tātou i ngā uara e mōhiotia ana:
– \( N_0 \) = 100,000
– \( k \) = 2% ia tau = 0.02
– \( t \) = 10 tau
Ka whakamahia e mātou te tātai tipu taupū:
\[ N(t) = N_0 \cdot e^{kt} \]
Monohia ngā uara e mōhiotia ana ki roto i te tātai:
\[ N(10) = 100,000 \cdot e^{0.02 \cdot 10} \]
\[ N(10) = 100,000 \cdot e^{0.2} \]
Ko te uara āwhiwhi o \( e^{0.2} \) he tata ki te 1.2214.
\[ N(10) = 100,000 \cdot 1.2214 \]
\[ N(10) = 122,140 \]
Nō reira, ko te taupori o te tāone i muri i ngā tau 10 e tata ana ki te 122,140 tāngata.
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Tauira Pātai 2: Uara Pūtea
Pātai
Ka tāpui koe i te $5,000 ki roto i tētahi pūkete e utu ana i te 5% huamoni ā-tau. He aha te uara o ō penapena i muri i ngā tau 15?
Te Whakatau
Tāutuhia ngā uara:
– \( N_0 \) = 5,000
– \( k \) = 5% ia tau = 0.05
– \( t \) = 15 tau
Mā te whakamahi i te tātai tipu taupū:
\[ N(t) = N_0 \cdot e^{kt} \]
Whakauruhia ngā uara e mōhiotia ana:
\[ N(15) = 5,000 \cdot e^{0.05 \cdot 15} \]
\[ N(15) = 5,000 \cdot e^{0.75} \]
Ko te whakatau tata \( e^{0.75} \) he tata ki te 2.117.
\[ N(15) = 5,000 \cdot 2.117 \]
\[ N(15) = 10,585 \]
Nō reira, ko te uara o ō penapena i muri i ngā tau 15 he tata ki te $10,585.
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Tauira 3: Te whakahekenga uara o te motuka
Pātai
Ka heke tere te uara o tētahi waka hou e $30,000 te utu, i te tere o te 12% ia tau. He aha te uara o te waka i muri i ngā tau e 5?
Te Whakatau
Tāutuhia ngā uara:
– \( N_0 \) = 30,000
– \( k \) = -12% ia tau = -0.12
– \( t \) = 5 tau
Mā te whakamahi i te tātai tipu taupū:
\[ N(t) = N_0 \cdot e^{kt} \]
Whakauruhia ngā uara e mōhiotia ana:
\[ N(5) = 30,000 \cdot e^{-0.12 \cdot 5} \]
\[ N(5) = 30,000 \cdot e^{-0.6} \]
Ko te whakatau tata \( e^{-0.6} \) he tata ki te 0.5488.
\[ N(5) = 30,000 \cdot 0.5488 \]
\[ N(5) = 16,464 \]
Nō reira, ko te uara o te motuka i muri i ngā tau e 5 he $16,464 pea.
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Tauira Pātai 4: Te horapa o ngā huaketo
Pātai
Kua pangia tētahi tāone e te huaketo, he 200 te taupori i te tīmatanga, ā, he tere tipu tere o te 15% ia rā. E hia te tapeke o ngā tāngata kua pangia i muri i ngā rā e 7?
Te Whakatau
Tāutuhia ngā uara:
– \( N_0 \) = 200
– \( k \) = 15% ia rā = 0.15
– \( t \) = 7 ngā rā
Mā te whakamahi i te tātai tipu taupū:
\[ N(t) = N_0 \cdot e^{kt} \]
Whakauruhia ngā uara e mōhiotia ana:
\[ N(7) = 200 \cdot e^{0.15 \cdot 7} \]
\[ N(7) = 200 \cdot e^{1.05} \]
Ko te whakatau tata \( e^{1.05} \) he tata ki te 2.857.
\[ N(7) = 200 \cdot 2.857 \]
\[ N(7) = 571.4 \]
Nō reira, ko te tapeke o te hunga i pangia i muri i ngā rā e 7 e tata ana ki te 571 tāngata.
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He tauira ēnei o ngā raruraru me ngā kōrero e pā ana ki te tipu taupū. He tino whai hua te mārama ki te whakaoti rapanga tipu taupū i roto i ngā mahi o te ao tūturu. Mēnā he nui te whakaharatau, ka taea te ngāwari te ako me te whakamahi i tēnei ariā ina hiahiatia.
