Answer:
The population will reach 100,000 in 2047.
The population will reach 200,000 in 2082.
Step-by-step explanation:
The compounded growth formula:
[tex]A=P(1+r)^t[/tex]
A= Population after t years
P= Initial amount of population
r= rate of growth
t= Time in years
Given that,
P=50,000, r=2% =0.02, A=100,000
[tex]\therefore 100,000=50,000(1+0.02)^t[/tex]
[tex]\Rightarrow 1.02^t=\frac{100,000}{50,000}[/tex]
[tex]\Rightarrow 1.02^t=2[/tex]
Taking ln both sides
[tex]\Rightarrow ln(1.02^t)=ln|2|[/tex]
[tex]\Rightarrow t\ ln(1.02)=ln|2|[/tex]
[tex]\Rightarrow t=\frac{ln|2|}{ ln(1.02)}[/tex]
[tex]\Rightarrow t\approx 35[/tex]
The population will reach 100,000 in (2012+35)=2047
P=50,000, r=2% =0.02, A=200,000
[tex]\therefore 200,000=50,000(1+0.02)^t[/tex]
[tex]\Rightarrow 1.02^t=\frac{200,000}{50,000}[/tex]
[tex]\Rightarrow 1.02^t=4[/tex]
Taking ln both sides
[tex]\Rightarrow ln(1.02^t)=ln|4|[/tex]
[tex]\Rightarrow t\ ln(1.02)=ln|4|[/tex]
[tex]\Rightarrow t=\frac{ln|4|}{ ln(1.02)}[/tex]
[tex]\Rightarrow t\approx 70[/tex]
The year it will reach 200,000 is (2012+70)=2082
