Ngā Tauira Pātai me te Kōrero mō te Pirau Taupūtanga
He āhuatanga taiao te pirau taupū e kitea ana i roto i ngā momo marautanga pēnei i te ahupūngao, te matū, te koiora, me te ōhanga. Hei tauira pāngarau, e whakaahua ana te pirau taupū i te tukanga e heke ai tētahi rahinga kua hoatu kia rite ki tōna rahinga o nāianei. I roto i te pāngarau, ka whai te pirau taupū i te āhua whānui:
\[ N(t) = N_0 e^{-\lambda t} \]
kāore i te mana:
– Ko te nui e toe ana i te wā \( t \) ko \( N(t) \),
– Ko te tau tīmatanga ko \( N_0 \),
– Ko te \( \lambda \) te pūmau pirau (e kiia ana ko te tere pirau),
– ko te wā ko te \( t \),
– Ko \( e \) te pūtake o te logarithm maori (tata ki te 2.718).
I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira o ngā raruraru pirau taupū me ā rātou otinga hei āwhina i a koe ki te mārama ake ki tēnei ariā.
Tauira Pātai 1: Te Pirau Irahiko
Pātai:
E rima tau te roa o te haurua-ora o tētahi matū irahiko. Mēnā i te tīmatanga he 100 karamu o taua matū, e hia te toenga i muri i ngā tau 15?
Kōrero:
Ka taea te whakatauira i te pirau irahiko mā te whakamahi i te tātai pirau taupū. Ko te haurua-ora (\( t_{1/2} \)) ko te wā e hiahiatia ana kia pirau te haurua o te nui o te rauemi irahiko. E mōhiotia ana ko \( t_{1/2} = 5 \) tau.
Tuatahi me kimi e tātou te pūmau pirau \( \lambda \) mā te whakamahi i te tātai:
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{5} \approx 0.1386 \text{ tau}^{-1} \]
Nō reira, ko te tātai pirau taupū ko:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(t) = 100 e^{-0.1386 \times 15} \]
Inaianei, ka tatauhia e mātou te uara:
\[ N(t) = 100 e^{-2.079} \]
\[ N(t) = 100 \whakareatia ki te 0.125 \]
\[ N(t) \tata ki te 12.5 \kuputuhi{ karamu} \]
Nō reira, i muri i ngā tau 15, e toe ana te 12.5 karamu o te rauemi irahiko.
Tauira 2: Te Pirau o te Pūnga
Pātai:
Ka whakaaetia kia tukuna he pūnga hiko me te utu tīmatanga \( Q_0 = 200 \text{ C} \) i roto i tētahi ara iahiko. Ko te pūmau wā ko \( \tau = 4 \text{ s} \). E hia te utu e toe ana i muri i te 10 hēkona?
Kōrero:
Mō te pirau o te utu pūnga, ko te tauira taupū e whakamahia ana ko:
\[ Q(t) = Q_0 e^{-t/\tau} \]
I homai \( Q_0 = 200 \text{ C} \) me \( \tau = 4 \text{ s} \). Me kimi e tātou \( Q(10) \):
\[ Q(10) = 200 e^{-10/4} \]
\[ Q(10) = 200 e^{-2.5} \]
Te tatau i ngā uara taupū:
\[ Q(10) = 200 \whakareatia ki te 0.0821 \]
\[ Q(10) \tata ki te 16.42 \kuputuhi{ C} \]
Nō reira, i muri i te 10 hēkona, ko te utu e toe ana i runga i te pūnga hiko he tata ki te 16.42 C.
Tauira Pātai 3: Pirau Matū
Pātai:
Ko te pūmau pirau o tētahi matū he \( \lambda = 0.05 \text{ rā}^{-1} \). Kia pēhea te roa ka heke te matū ki te 25% o tōna rahinga taketake?
Kōrero:
Ka tīmata tātou me te tātai whānui mō te pirau taupū:
\[ N(t) = N_0 e^{-\lambda t} \]
E hiahia ana mātou kia 25% o te N(t) o \( N_0 \), kia:
\[ 0.25 N_0 = N_0 e^{-0.05 t} \]
Te whakakore i te \( N_0 \) mai i ngā taha e rua:
\[ 0.25 = e^{-0.05 t} \]
Te whakamahi i ngā taupūnga maori hei whakaoti rapanga taupūnga:
\[ \ln 0.25 = -0.05 t \]
\[ -1.3863 = -0.05 t \]
Te whakaoti rapanga mō \( t \):
\[ t = \frac{1.3863}{0.05} \]
\[ t \tata ki te 27.726 \kuputuhi{ ngā rā} \]
Nō reira, ko te wā e hiahiatia ana kia heke ai te matū ki te 25% o tōna rahinga tīmatanga he tata ki te 27.726 ngā rā.
Tauira Pātai 4: Te Pirau o te Taupori Kitakita
Pātai:
He tere te heke o te taupori huakita, ā, i muri i ngā hāora e 3, ka haurua te taupori i tōna tau tīmatanga. Mēnā he 8000 ngā huakita i te taupori tīmatanga, e hia ngā huakita e toe ana i muri i ngā hāora e 9?
Kōrero:
E mōhiotia ana ko te haurua-ora \( t_{1/2} = 3 \) hāora. Tuatahi ka kitea e tātou te pūmau pirau \( \lambda \):
\[ \lambda = \frac{\ln 2}{t_{1/2}} \]
\[ \lambda = \frac{\ln 2}{3} \approx 0.231 \text{ haora}^{-1} \]
Whai muri i tēnā, ka whakamahia e mātou te tātai pirau taupū:
\[ N(t) = N_0 e^{-\lambda t} \]
\[ N(9) = 8000 e^{-0.231 \times 9} \]
Te tatau i ngā uara taupū:
\[ N(9) = 8000 e^{-2.079} \]
\[ N(9) = 8000 \whakareatia ki te 0.125 \]
\[ N(9) \tata ki te 1000 \]
Nō reira, i muri i ngā hāora e 9, ka toe tonu mai te 1000 huakita.
Whakamutunga
Mā te tauira pirau taupū ka whai hua te huarahi ki te whakaoti rapanga e pā ana ki ngā tukanga pirau i roto i ngā momo tono pūtaiao me te hangarau. Mā te mārama ki ngā ariā taketake pēnei i ngā pūmau pirau, ngā haurua-ora, me te whakamahinga o ngā tātai taupū, ka taea e tātou te tatau i te huringa o tētahi rahinga i roto i te wā me te ngāwari. Mā ngā raruraru mahi kua kōrerotia i runga ake nei ka āwhina i a tātou ki te mārama me te whakamahi i te ariā o te pirau taupū ki ngā āhuatanga uaua ake.
