The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. =>
dx/dt=kt
Integrate, x(t)=kt^2/2+C ................(0)
Substitute x(3)=400 => k(3^2)/2+C=400 => 4.5k+C=400 .........(1)
Substitute x(10)=2000 => k(10^2))/2+C=2000 => 50k+C=2000 .........(2)
(2)-(1) => 45.5k=1600 => k=35.165 .............(3)
substitute (3) in (1)
4.5(35.165)+C=400 => C=241.76 .................(4)
Substitute (3) and (4) in (0) to get
x(t)=kt^2/2+C=17.582t^2+241.76 .............(5)
So at t=0, x(0)=241.7 or 242 (to the nearest bacterium)
