Question

A species of animal is discovered on an island. Suppose that the population size P(t) of the species can be modeled by the following function, where time t is measured in years.
P(t) =800/1+ 3e^-0.29t.
Find the initial population size of the species and the population size after 10 years. Round your answers to the nearest whole number as necessary.

Answer 1

Answer:

The initial population is 200

The population after 10 years is 687

Step-by-step explanation:

Given

$P(t) =\frac{800}{1+ 3e^{-0.29t.}}$

Solving (a): The initial population

Here:

$t = 0$

Substitute 0 for t in P(t)

$P(0) =\frac{800}{1+ 3e^{-0.29*0}}$

$P(0) =\frac{800}{1+ 3e^{0}}$

$P(0) =\frac{800}{1+ 3*1}$

$P(0) =\frac{800}{1+ 3}$

$P(0) =\frac{800}{4}$

$P(0) =200$

The initial population is 200

Solving (b): Population after 10 years.

Here

$t = 10$

Substitute 10 for t in P(t)

$P(10) =\frac{800}{1+ 3e^{-0.29*10}}$

$P(10) =\frac{800}{1+ 3e^{-2.9}}$

$P(10) =\frac{800}{1+ 3*0.055}$

$P(10) =\frac{800}{1+ 0.165}$

$P(10) =\frac{800}{1.165}$

$P(10) =687$ -- approximated.

The population after 10 years is 687