Rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant and it is found that the number doubles in 6 hours. The number of bacteria becomes times at the end of 18 hours.

Step-by-step solution

Let P₀ be the initial population and let the population after t years be P . Then, aligned & dP dt = kP , where k >0 & dP P = kdt aligned Integrating on both sides, we get aligned & P = kt + c & When t =0, P = P ₀ & P ₀=0+ c & c= P ₀ & P = kt + P ₀ & P P ₀ = kt & aligned & P = kt + P ₀ & P P ₀ = kt & When t =6 hrs, P =2 P ₀ aligned & 2 P ₀ P ₀ =6 k & k = 2 6 & P P₀ = 2 6 t & When t =18 hrs , we have & P P₀ = 2 6 18 & =3 2 & P P₀ = 8 & P =8 P ₀ & aligned