Question

The population of a county is growing at a rate of 9% per year, compounded continuously. How many years will it take for the population to quadruple according to the exponential growth function?

Answer 1

Answer:

$t=15.4\ years$

Step-by-step explanation:

The  exponential growth function compounded continuously is equal to

$A=P(e)^{rt}$  

where  

A is the final population  

P is the initial population  

r is the rate of growth in decimal  

t is Number of years

e is the mathematical constant number

we have  

$A=4x\\P=x\\ r=9\%=9/100=0.09$  

substitute in the function above

$4x=x(e)^{0.09t}$    

simplify

$4=(e)^{0.09t}$

Take natural log of both sides

$ln(4)=ln[(e)^{0.09t}]$

$ln(4)=0.09t(ln(e))$

$ln(e)=1$

$ln(4)=0.09t$

$t=ln(4)/0.09$

$t=15.4\ years$