Question

Write a function f(t), the models the number of foxes in the population at t years after 2012, assuming a continuous growth rate.. f(t)=what is the fox population predicted to be in 2020?

Answer 1

It is given that the growth rate s 4%

The population in 2012 is 20040.

Recall the formula for the growth rate.

$Percent\text{ rate=}\frac{\frac{\text{present populaiton-past poulation}}{\text{past pulation}}\times100}{\text{ the number of years}}$

$Percent\text{ rate=}\frac{\frac{f(t)-20040}{20040}\times100}{t}$

Substitute per cent rate =4 %, Present population =f(t) , the number of years and

Past population=20040 as follows:

$4=\frac{\frac{f(t)-20040}{20040}\times100}{t}$

$4t=\frac{f(t)-20040}{20040}\times100$

$\frac{4t}{100}=\frac{f(t)-20040}{20040}$

$\frac{4t}{100}\times20040=f(t)-20040$

$0.04t\times20040+20040=f(t)$

$20040(0.04t+1)=f(t)$

Hence the number of foxes in the population at t years after 20212 is

$f\mleft(t\mright)=(0.04t+1)20040$

In the year 2020

The number of years after 2012 is =2020-2012=8

Substitute t=8 in the model f(t), we get

$f\mleft(8\mright)=(0.04\times8+1)20040$

$f\mleft(8\mright)=(1.32)20040$

$f(8)=26452.8$

The number of foxes in the population predicted to be in 2020 is 26452.