The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. after 3 hours it is observed that 400 bacteria are present. after 10 hours 2000 bacteria are present. what was the initial number of bacteria? (round your answer to the nearest integer.)

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)