exponential function real world

Question

a new prototype of nanobots self replicate continually at a rate such that their population doubles every three hours. If we initially observe that there are 4000 nanobots, write an equation for the population as a function of time (in hours), and find how long it will take for the population to reach 35000

Answer 1

P(t) = population as function of time

 

P(t) = P(0)e^kt with t in hours

 

P(t+3) = 2P(t)

 

P(0)e^k(t+3) = 2P(0)e^kt

 

e^3k = 2

 

k = ln(2) / 3 per hour = 0.231049 per hour

 

P(0) = 4000

 

P(t) = 4000e^0.231049t with t in hours

 

Time to reach 35000:

 

35000 = 4000e^0.231049t

 

t = [ln(35000 / 4000)] / (0.231049) hours = 9.38785 hours