The count in a bateria culture was 800 after 15 minutes and 1600 after 40 minutes. assuming the count grows exponentially,

what was the initial size of the culture?

find the doubling period?

find the population after 65 minutes?

when will the population reach 1400?

1) bacteria size = b(initial) * e^(r * t) 

initial size = 300 
final size = 1600.
 
Substituting the values into the main equation:

1600 = 300 * e^(r * 35 - 20) 
solving for 'r'.

1600 = 300 * e^(r * 15) 

1600/300 = e^(r * 15) 
taking natural log on both sides to eliminate the power

ln(16/3) = ln[e^(r * 15)] 

ln(16/3) = r * 15 

r  = ln(16/3) / 15
   = 0.11159843
   = 11.16% 

To find the initial bacteria size,  
Given: t = 20 

300 = b * e^(0.1116 * 20) 

Solving for b:

300 = b * e^2.232 

b = 300/e^2.232 

b = 32 

2) The doubling period = 70% / r (in percent)
                                     = 70% / 11.16%per min
                                     = 6.272 minutes 

3) Population after 65 minutes = 32 * e^(0.1116 * 65) 
X = 32 * e^7.254 
X = 631

4) When will the population reach 1400

14000 = 32 * e^(0.1116 * t) 
solving for t 
ln(14000/32) = ln(e^(.1116 * t) 
ln(437.5) =0.1116 * t 
t = ln(437.5) /0.1116 = 54.5 minutes