Question

The population (in millions) of a certain island can be approximated by the function p(x)=50(1.05)^x, where x is the number of years since 2000. in which year will the population reach 200 million? hint: an answer such as 2002.4 would represent 2002. a)2028

b)2015
c)2002
d)2066
please help

Answer 1

The answer is A)2028 - just took the test

Answer 2

Answer: a)2028

Explanation:

Given: The population (in millions) of a certain island can be approximated by the function
`p(x)=50(1.05)^x` , 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

`200=50(1.05)^x\\\\\Rightarrow(1.05)^x=4`

Taking natural log on both the sides , we get

`\ln((1.05)^x=\ln(4)\\\\\Rightarrow\ x(\ln(1.05))=\ln4\\\\\Rightarrow\ x=(\ln4)/(\ln(1.05))=(1.38629)/(0.04879)\\\\\Rightarrow\ x=28.4134043861\approx28\text{ years}`

The year will be
`2000+28=2028`

Hence, in 2028 the population will reach 200 million.