Question

During a gold rush in the 1800's, the population of a boom town increased exponentially. The population can be modeled by the function A(t)=A₀e^rt, where t is the time in months after gold is discovered. If the initial population is 4 people, and there are 8 people after 2 months, how long does it take for the population to reach 32 people? (Give an exact answer, not a decimal approximation.)

Answer 1

A(t) = A​​​​​​0 e​​​​​​rt

t = time period(in months)

# at = 0 (initially) , A(0) = 4

So, 4 = A​​​​​​0 e​​​​​​0   =>  ​​​​​​​​​​​​A0 = 4

Then , A(t) = 4 e​​​​​​rt

# Now , at t = 2 , A(2) = 8

=> 8 = 4 e​​​​​​r(2)  

=> 2 = e​​​​​​2r

Taking logarithm both sides

=> ln(2) = 2r

=> r = ln(2)/2

Therefore ,

# To reach A(t) = 32

=>  

=>  

Taking logrithm both sides :

=>  

=>

=>

So at t = 6 months , population will be 32 people.