Lionfish are considered an invasive species, with an annual growth rate of 69%. A scientist estimates there are 9,000 lionfish in a certain bay after the first year.

The general equation of the growth is:

[tex]P(t)=P_0\cdot(1+r)^t[/tex]

Given rate = r = 69% = 0.69

After 1 year, P = 9000

Substitute to find the initial number of Lionfish

So,

[tex]\begin{gathered} 9000=P_0\cdot(1+0.69)^1 \\ 9000=P_0\cdot1.69 \\ P_0=\frac{9000}{1.69}\approx5325 \end{gathered}[/tex]

Part (A), we will write an explicit formula f(n) that represents the number of lionfish after n years

so, the formula will be:

[tex]f(n)=5325\cdot1.69^n[/tex]

Part (B): we will find the number of lionfish after 6 years

so, substitute with n = 6 into the equation of part (a)

[tex]f(6)=5325\cdot1.69^6=124,073[/tex]

So, after 6 years, the number of lionfish = 124,073

Part (C): The scientists remove 1400 fish per year after the first year

So, we the number of lionfish:

[tex]9000-1400=7600[/tex]

Then after 2 years, the number of lionfish

[tex]7600\cdot1.69-1400[/tex]

After 3 years:

[tex]\begin{gathered} (7600\cdot1.69-1400)\cdot1.69-1400 \\ =7600\cdot1.69^2-1400\cdot1.69-1400 \\ =7600\cdot1.69^2-1400\cdot(1+1.69) \end{gathered}[/tex]

So, after (n) years:

[tex]7600\cdot1.69^{n-1}-1400\cdot(1+1.69)^{n-2}^{}[/tex]