Calculus kwa Kiswahili

Calculus kwa Kiswahili

Teknocrat

Teknocrat

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Kijiji cha Chato kina kiwango cha kuzaliwa na kiwango cha kifo ambacho kinalingana na jumla ya watu.

Idadi ya watu huongezeka kila baada ya miaka 30.

Wakati wa muda wa t = 0, idadi ya watu ni 1000.

Tafuta idadi ya watu baada ya miaka 40 na wakati uliochukuliwa kwa idadi ya watu kuongezeka mara kumi?
 
Teknologist said:
Kijiji cha Chato kina kiwango cha kuzaliwa na kiwango cha kifo ambacho kinalingana na jumla ya watu.

Idadi ya watu huongezeka kila baada ya miaka 30.

Wakati wa muda wa t = 0, idadi ya watu ni 1000.

Tafuta idadi ya watu baada ya miaka 40 na wakati uliochukuliwa kwa idadi ya watu kuongezeka mara kumi?
Click to expand...
?????????!!!!!!!!!!!!!
 
Makanyaga said:
Badilisha jina la kijiji kwanza. Zaidi inaonekana kama swali limekosewa kwenye statement hii:
Idadi ya watu huongezeka kila baada ya miaka 30.
Click to expand...
OK , corrected...
Kijiji cha Kolomije kina kiwango cha kuzaliwa na kiwango cha kifo ambacho kinalingana na jumla ya watu.

Idadi ya watu huongezeka mara mbili kila baada ya miaka 30.

Wakati wa muda wa t = 0, idadi ya watu ni 1000.

Tafuta idadi ya watu baada ya miaka 40 na wakati uliochukuliwa kwa idadi ya watu kuongezeka mara kumi?
 
Teknologist said:
OK , corrected...
Kijiji cha Kolomije kina kiwango cha kuzaliwa na kiwango cha kifo ambacho kinalingana na jumla ya watu.

Idadi ya watu huongezeka mara mbili kila baada ya miaka 30.

Wakati wa muda wa t = 0, idadi ya watu ni 1000.

Tafuta idadi ya watu baada ya miaka 40 na wakati uliochukuliwa kwa idadi ya watu kuongezeka mara kumi?
Click to expand...
Andika version ya kiingereza
 
Teknologist said:
Kijiji cha Kolomije kina kiwango cha kuzaliwa na kiwango cha kifo ambacho kinalingana na jumla ya watu.
Click to expand...


Unatafuta nini iwapo idadi ya vifo na uzazi vinalingana??, idadi ya watu ya hicho kijiji itabakia kuwa CONSTANT.
 

Makanyaga​

Mzee acha longolongo, tulidhani mnasapoti kila ngazi ya elimu iwe kwa kiswahili....

Mokaze​

Labda hata wewe utaeelewa zaidi kwa lugha ya kigeni...

Haya Swali hili hapa

An isolated colony of humans has a birth rate and a death rate that are proportional to the total population. The population doubles every 30 years.
At time t = 0, the population is 1000 humans. Calculate the population after 40 years and the time taken for the population to increase tenfold.
 
Teknologist said:

Makanyaga​

Mzee acha longolongo, tulidhani mnasapoti kila ngazi ya elimu iwe kwa kiswahili....

Mokaze​

Labda hata wewe utaeelewa zaidi kwa lugha ya kigeni...

Haya Swali hili hapa

An isolated colony of humans has a birth rate and a death rate that are proportional to the total population. The population doubles every 30 years.
At time t = 0, the population is 1000 humans. Calculate the population after 40 years and the time taken for the population to increase tenfold.
Click to expand...
Ulitupotosha. Neno propotional halitafisiriki kama usawa ama lingana bali utafisirika kama wiana.

Kuwiana siyo sawa na neno kuwasawa. Ndiyo maana mtaalamu wa kalikulasi hapo juu kasema unasumbuka na nini wakati suala ni konsitanti.
 
Lipeleke swali kwenye lugha ya malkia nikusaidie
 
Mbona swali hili hapa nilishaliweka.....

An isolated colony of humans has a birth rate and a death rate that are proportional to the total population. The population doubles every 30 years.
At time t = 0, the population is 1000 humans. Calculate the population after 40 years and the time taken for the population to increase tenfold.
 
P(t)=P.e^kt..
population double every 30 years
.....................
......................
.....................
....................
 
Teknologist said:

Makanyaga​

Mzee acha longolongo, tulidhani mnasapoti kila ngazi ya elimu iwe kwa kiswahili....

Mokaze​

Labda hata wewe utaeelewa zaidi kwa lugha ya kigeni...

Haya Swali hili hapa

An isolated colony of humans has a birth rate and a death rate that are proportional to the total population. The population doubles every 30 years.
At time t = 0, the population is 1000 humans. Calculate the population after 40 years and the time taken for the population to increase tenfold.
Click to expand...


The population after 40 yrs is 2,520 humans, and the time for the population to be tenfold is 99.6 years close to 100years. 😀,

Sikuona kama ulinitag siku ile.
 
Mokaze said:
The population after 40 yrs is 2,520 humans, and the time for the population to be tenfold is 99.6 years close to 100years. 😀,

Sikuona kama ulinitag siku ile.
Click to expand...
Aaaaaa..
Weka minofu tuione mzee wa maths...😀
P=population at time t
c=initial population
k=rate of growth

dP/dt=kp...
dP=kpdt...
dP/p=kdt...
Integrate both sides
∫dP/p=∫kdt
In(P)=kt+c
take out (In)
P'=e^kt+c
P=e^c+e^kt
e^c=c
P=c.e^kt..............general formula of expontential growth/and or decay...

Doubling time...
P=2c
with P=c.e^kt
2c=c.e^kt
cancelling c..
2=e^kt.........doubling time formula

Lugha kawaida inawachanganya sana wanafunzi wengi bongo japo wanajua sana hesabu...
wanatakiwa kujua kubalisha statement kuwa hesabu na hapo ndio ugumu....
Lakini ukiwaletea za 4x^3+2x^2+3x+5=0.......wanazipopoa hasa....
 
Teknologist said:
Aaaaaa..
Weka minofu tuione mzee wa maths...😀
P=population at time t
c=initial population
k=rate of growth

dP/dt=kp...
dP=kpdt...
dP/p=kdt...
Integrate both sides
∫dP/p=∫kdt
In(P)=kt+c
take out (In)
P'=e^kt+c
P=e^c+e^kt
e^c=c
P=c.e^kt..............general formula of expontential growth/and or decay...

Doubling time...
P=2c
with P=c.e^kt
2c=c.e^kt
cancelling c..
2=e^kt.........doubling time formula

Lugha kawaida inawachanganya sana wanafunzi wengi bongo japo wanajua sana hesabu...
wanatakiwa kujua kubalisha statement kuwa hesabu na hapo ndio ugumu....
Lakini ukiwaletea za 4x^3+2x^2+3x+5=0.......wanazipopoa hasa....
Click to expand...



For the case of the Qns, let me put mnofu.

Starting from; lnP=kt+C ----(1)

At t=0, P=1000, plugging into eqn 1

➡ln1000=0×k+C, therefore, c=ln1000,

Plugging c=ln1000 into eqn 1 to have a specific eqn,

➡lnp=kt+ln1000 -----(2)

The given condition is; for 30 years the population doubles, that is starting t=0 after t=30 years the population will be 1000×2=2000 humans, plugging the stuffs into eqn 2, finding k,

➡In2000= 30k+ln1000, rearranging for k;

In2000-ln1000=30k
ln(2000/1000)=30k
Therefore k=(ln2)/30

Now the eqn2 becomes;

ln p=(ln2)/30t+ln 1000
lnp=ln2^0.033333t +ln1000,
lnp=ln(2^0.03333t×1000), removing ln frim both sides of the equation

➡P=2^0.033333t×1000, that is the equation to be used to solve the Qns:

1----The population p after 40 years.
2----The time t for the population to become tenfold.
 
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