find the population of a city at any time t, given that the rate of increase of population is proportional to the population at that instant and that in a period of 40 years the population increased from 30,000 to 40,000
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Answer:
Let P be the population at any time t. It is given that
dt
dP
∞P
⇒
dt
dP
=λP, where λ is a constant of proportionally
⇒
P
dP
=λdt
⇒logP=λt+logC ……(i)
Initially i.e., at t=1990, P=200000 and at t=2000, P=250000
∴log200000=1990λ+logC ……..(i) and, log250000=2000λ+logC ……..(ii)
⇒log200000−log250000=10λ [On subtracting (ii) from (i)]
⇒λ=
10
1
log(
5
4
)
Putting λ=
10
1
log(
5
4
) in (i), we get
log 200000=1990×
10
1
log(
5
4
)+logC
⇒log 200000=199log
5
4
+logC
⇒logC=log200000−199log
5
4
Putting λ=
10
1
log
5
4
,logC=log200000−199 log
5
4
and t=2010 in (i), we get
logP=(
10
1
log
5
4
)2010+log200000−199log
5
4
⇒logP=log(
5
4
)
201
+log(200000×(
4
5
)
199
)
⇒P=(
5
4
)
201
×200000×(
4
5
)
199
=(
4
5
)
2
×200000=
16
25
×200000=312500.
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