MAT3714 ASSIGNMENT 2 2025

MAT3714
ASSIGNMENT 2
2025

, TUTORIAL 5


QUESTION 1




SOLUTION:


The population increases at the rate proportional to the number of people
present at time t.


dP
∝P
dt
dP
= kP
dt
1
∫ dP = ∫ k dt
P
ln(P) = kt + C

P(t) = ekt+C

P(t) = ekt eC ∴ Let: A = eC
P(t) = Aekt


∴ P(t) = P0

Aek(0) = P0
Ae0 = P0 ∴ e0 = 1
A = P0


P(t) = Aekt
P(t) = P0 ekt

,The population doubled in 5 years:


P(5) = 2P0

P0 ek(5) = 2P0
2P0
e5k =
P0
e5k = 2
5k = ln(2)
ln(2)
k=
5


P(t) = P0 ekt
ln(2)
( )t
P(t) = P0 e 5




i).


For the population to triple:


P(t) = 3P0
ln(2)
( )t
P0 e 5 = 3P0
ln(2)
( )t
e 5 =3
ln(2)
t = ln(3)
5
5 ln(3)
t=
ln(2)
t = 7.93 years


It will take about 7.23 years for the population to triple.

, ii).


For the population to quadruple:


P(t) = 4P0
ln(2)
( )t
P0 e 5 = 4P0
ln(2)
( )t
e 5 =4
ln(2)
t = ln(4)
5
5 ln(4)
t=
ln(2)
t = 10 years


It will take about 10 years for the population to quadruple.




QUESTION 5




SOLUTION:


dP
− ∝P
dt
dP
= −kP
dt
1
∫ dP = ∫ −k dt
P
ln(P) = −kt + C