Answer: a) k= 0.025
The exponential growth function in time t is given by :-
[tex]P=66000e^{0.025t} [/tex],
b) In year 2033 the population will reach to 176,000.
Step-by-step explanation:
The exponential growth function in time t is given by :-
[tex]P=P_0e^{kt} [/tex], where k is the rate of growth , [tex]P_0[/tex] is the initial population.
Given : In 2016 , the initial population of town = [tex]66,000[/tex]
The rate of growth per year=[tex]k=2.5\%[/tex]
Which can be written as
[tex]k=0.025[/tex]
Let t be the number of years since 2016 to take population reach 176,000.
Then , the required equation will be :-
[tex]176000=66000e^{0.025t}\\\\\Rightarrow\ e^{0.025t}=\dfrac{176}{66} \\\\\Rightarrow\ e^{0.025t}=2.67[/tex]
Taking log on both sides , we get
[tex]0.025t=\log(2.67)\\\\\Rightarrow\ 0.025t=0.426511261365\\\\\Rightarrow\ t=17.0604504\approx17[/tex]
Thus it will take 17 years since 2016 to reach population 176,000.
Hence, In year 2033 the population will reach to 176,000.
