Natural growth and decay rate of a bacteria culture

Question

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 $10,000$?

Answer 1

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 $k$.