A species of fish was added to a lake. The population size P() of this species can be modeled by the following function, where t is the number of years from the
time the species was added to the lake.
P(t)=2000/1+3e^-0.34t
Find the initial population size of the species and the population size after 9 years.
Round your answers to the nearest whole number as necessary.
Initial population size: ?
Population size after 9 years: ?

We usually represent real situations as mathematical functions or rules that express the dependency of one variable quantity P with another variable quantity t.

The population size of a species of fish P(x) is modeled by the following function:

[tex]\displaystyle P(t)=\frac{3000}{1+3e^{-0.34t}}[/tex]

Where t is the number of years elapsed since the species was added to the lake.

The initial population size can be found by substituting t for 0:

[tex]\displaystyle P(0)=\frac{2000}{1+3e^{-0.34*0}}[/tex]

[tex]\displaystyle P(0)=\frac{2000}{1+3*1}[/tex]

[tex]\displaystyle P(0)=\frac{2000}{4}[/tex]

P(0)=500

The initial population size is 500 fish

The population size after t=9 years is:

[tex]\displaystyle P(9)=\frac{2000}{1+3e^{-0.34*9}}[/tex]

[tex]\displaystyle P(9)=\frac{2000}{1+3e^{-3.06}}[/tex]

[tex]\displaystyle P(9)=\frac{2000}{1.047}[/tex]

[tex]P(9)\approx 1910[/tex]

Population size after 9 years: 1910 fish