Answer: a)2028
Step-by-step explanation:
Given: The population (in millions) of a certain island can be approximated by the function [tex]p(x)=50(1.05)^x[/tex] , where x is the number of years since 2000.
To find the year in in which year will the population reach 200 million, substitute p(x)=200, we get
[tex]200=50(1.05)^x\\\\\Rightarrow(1.05)^x=4[/tex]
Taking natural log on both the sides , we get
[tex]\ln((1.05)^x=\ln(4)\\\\\Rightarrow\ x(\ln(1.05))=\ln4\\\\\Rightarrow\ x=\dfrac{\ln4}{\ln(1.05})=\dfrac{1.38629}{0.04879}\\\\\Rightarrow\ x=28.4134043861\approx28\text{ years}[/tex]
The year will be [tex]2000+28=2028[/tex]
Hence, in 2028 the population will reach 200 million.
