Lipotso tsa Mehlala le Puisano ea Mesebetsi ea Exponential
Mesebetsi ea exponential ke mohopolo oa motheo lipalo, o akaretsang phetoho ea exponential, kholo ea exponential le ho bola ha exponential. Kutloisiso e felletseng ea mesebetsi ena e na le lits'ebetso tse ngata tse sebetsang bophelong ba 'nete, ho tloha k'hemistri le fisiks ho ea ho baeloji le moruo. Sengoloa sena se tla hlahloba mehlala e 'maloa ea mesebetsi ea exponential le litharollo tsa eona ho thusa ho ntlafatsa kutloisiso ea hau ea sehlooho sena.
Selelekela sa Mesebetsi ea Exponential
Mosebetsi wa exponential o na le sebopeho se akaretsang \( y = a \cdot b^x \), moo:
– \( y \) ke boleng ba tshebetso
– \( a \) ke ntho e sa fetoheng
– \( b \) ke motheo wa exponential
– \( x \) ke phetoho e ikemetseng
Ka tlwaelo, haeba \( b > 1 \), mosebetsi o ba le kgolo ya exponential, mme haeba \( 0 < b < 1 \), mosebetsi o ba le bothata ba exponential. Mathata a Mohlala ka Mesebetsi ya Exponential Mona ke mehlala ya mathata ho bontsha tshebediso ya mesebetsi ya exponential le puisano ya bona e qaqileng. Mohlala Bothata ba 1: Kgolo ya Baahi Bothata: Palo ya baktheria e na le diphedi tse 500 mme e ata ka sekgahla se ka etswang mohlala ke mosebetsi wa exponential \( P(t) = 500 \cdot 2^t \), moo \(t \) e lekanngwang ka dihora. Palo ya baktheria ke efe kamora dihora tse 5?
Puisano: Bothateng bona, rea tseba: - Palo ea batho ba qalang, \( P_0 = 500 \) - \( b = 2 \) - \( t = 5 \) Re hloka feela ho sebelisa boleng ba \( t \) mosebetsing o fanoeng oa exponential: \[ P(5) = 500 \cdot 2^5 \] Ho bala \( 2^5 \): \[ 2^5 = 32 \] Joale, atisa ka palo ea batho ba qalang: \[ P(5) = 500 \cdot 32 = 16000 \] Kahoo, palo ea batho ba bakang baktheria ka mor'a lihora tse 5 ke lintho tse phelang tse 16.000. Mohlala Bothata ba 2: Ho Bola ha Mahlaseli Bothata: Sampole ea mahlaseli e na le ligrama tse 200 tsa ntho e nang le halofo ea bophelo ba lihora tse 3. Mosebetsi wa exponential o hlalosang bongata ba ntho e setseng kamora dihora tse \( t \) ke \( N(t) = 200 \cdot \left(\frac{1}{2}\right)^{t/3} \). Ke ntho e kae e setseng kamora dihora tse 9? Tharollo: Bothateng bona, re a tseba: - Boima ba pele, \( N_0 = 200 \) digrama - Motheo wa exponent, \( b = \frac{1}{2} \) - \( t = 9 \) Re nkela boleng ba \( t = 9 \) sebakeng sa mosebetsi wa exponential: \[ N(9) = 200 \cdot \left(\frac{1}{2}\right)^{9/3} \] Nolofatsa di-exponents: \[ 9/3 = 3 \] Kahoo mosebetsi o fetoha: \[ N(9) = 200 \cdot \left(\frac{1}{2}\right)^3 \] Ho bala \( \left( \frac{1}{2} \right)^3 \): \[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \] Jwale, atisa ka boima ba pele: \[ N(9) = 200 \cdot \frac{1}{8} = 25 \] Kahoo, bongata ba ntho e setseng kamora dihora tse 9 ke digrama tse 25. Mohlala Bothata ba 3: Bothata ba Kgolo ya Moruo: Naha e na le kgolo ya moruo ya 4% ka selemo, e ka etswang mohlala ke mosebetsi wa exponential \( G(t) = G_0 \cdot (1.04)^t \), moo \( G_0 \) e leng GDP ya pele mme \(t \) e le nako ka dilemo. Haeba GDP ya pele e le \( G_0 = 1.000.000 \), GDP ya yona e tla ba eng kamora dilemo tse 7? Tharollo: E fanoe: - GDP ea pele, \( G_0 = 1.000.000 \) - Sekhahla sa kholo, \( b = 1.04 \) - \( t = 7 \) Re nkela boleng ba \( t = 7 \) sebaka sa mosebetsi oa exponential: \[ G(7) = 1.000.000 \cdot (1.04)^7 \] Ho bala \( (1.04)^7 \): \[ (1.04)^7 \hoo e ka bang 1.316074 \] Joale, atisa ka GDP ea pele: \[ G(7) = 1.000.000 \cdot 1.316074 \hoo e ka bang 1.316.074 \] Kahoo, GDP kamora dilemo tse 7 e hakanngwa ho ba hoo e ka bang 1.316.074. Mohlala Bothata ba 4: Boleng ba Matsete Bothata: Letsete la pele la 20.000 ka sekgahla sa tswala sa selemo sa 5% se sebediswang ka ho kopanya selemo le selemo le ka etswa mohlala ke mosebetsi \( A(t) = 20000 \cdot (1+0.05)^t \), moo \( A(t) \) e leng boleng bohle ba letsete kamora dilemo \(t \). Bala boleng ba letsete kamora dilemo tse 10. Tharollo: E fanoe: - Matsete a pele, \( A_0 = 20000 \) - Sekgahla sa tswala sa selemo le selemo, \( b = 1.05 \) - \( t = 10 \) Re nkela boleng ba \( t = 10 \) sebaka mosebetsing wa exponential: \[ A(10) = 20000 \cdot (1.05)^{10} \] Ho bala \( (1.05)^{10} \): \[ (1.05)^{10} \hoo e ka bang 1.62889 \] Jwale, atisa ka matsete a pele: \[ A(10) = 20000 \cdot 1.62889 \hoo e ka bang 32.577,80 \] Kahoo, boleng ba matsete kamora dilemo tse 10 ke hoo e ka bang 32.577,80. Mesebetsi ya exponential ke disebediswa tse matla dipalong tse nang le mefuta e mengata ya ditshebediso tse sebetsang. Ho tloha kgolong ya baahi ho isa ho bola ha mahlaseli a kotsi le kgolo ya moruo, ho utlwisisa le ho sebedisa mesebetsi ya exponential ho bohlokwa. Ho buisana ka mehlala e kang e ka hodimo ho thusa ho hlakisa mehopolo le ho ntlafatsa bokgoni ba ho rarolla mathata. Tswela pele ho ikwetlisa le ho hlahloba ditshebediso tse fapaneng tsa mesebetsi ya exponential ho tebisa kutlwisiso ya hao.