The population increases by 10x after 16.6 years
Step-by-step explanation:
The population in this problem doubles every 5 years; this means that we can write an expression for the population after n years as follows:
[tex]p(n)=(2)^{\frac{n}{5}}p_0[/tex]
Where
[tex]p_0[/tex] is the initial population at n = 0
We can verify that with this expression, we get correctly that the population doubles after 5 years:
[tex]p(5)=(2)^{\frac{5}{5}}=2p_0[/tex]
And it keeps doubling after every 5 years:
[tex]p(10)=2^{\frac{10}{5}}p_0 = 4 p_0[/tex]
And so on.
Now we want to find the number of years n after which the population has increased by 10x, so that
[tex]p(n)=10p_0[/tex]
We can write therefore
[tex](2)^{\frac{n}{5}}p_0 = 10p_0[/tex]
And solving for n:
[tex](2)^{\frac{n}{5}}=10\\\frac{n}{5}=log_2(10)\\n=5 log_2(10)=16.6[/tex]
Therefore, the population increases by 10x after 16.6 years.
Learn more about population growth:
brainly.com/question/10689103
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