# Natural growth and decay rate of a bacteria culture

A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $420$.

(a) Find an expression for the number of bacteria after $t$ hours.

(b) Find the number of bacteria after $3$ hours.

(c) Find the rate of growth after $3$ hours.

(d) When will the population reach $10,000$?

• In which part did you run into a problem? – Librecoin Nov 8 '13 at 20:21
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Hint: Note that the number of the bacteria $N$ increases with respect to time $t$ which suggests using the differential equation
$$\frac{dN(t)}{dt}=k N(t),\quad k>0.$$
Solve the ode and use the initial condition $N(0)=100$ to find the constant of integration $C$ which resulted from solving the ode. Then advance to find $k$.