Answer to Question #215336 in Differential Equations for Zerin

Question

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time .After hours it is observed that there are After hours there are bacteria present. bacteria present. Find the initial number of bacteria .

Answer 1

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 there are 400 bacteria present. After 10 hours there are 2000 bacteria present. What is the initial number of bacteria?

$$\dfrac{dN}{dt}=kN$$

$$\int\dfrac{dN}{N}=\int k dt$$

$$\ln|N|=kt+\ln N_0$$

$$N(t)=N_0e^{kt}$$

$$N(3)=400=>400=N_0e^{k(3)}$$

$$N(10)=2000=>2000=N_0e^{k(10)}$$

$$e^{7k}=5$$

$$e^k=5^{1/7}$$

$$N_0=400\cdot(5)^{-3/7}$$

$$N_0=201$$