The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0 . The number of bacteria is increased by 20 % in 2 hours. If the population of bacteria is 2000 after k log e 6 5 hours, then k log e 2 2 is equal to:

Step-by-step solution

Given d B d t ∝ B ⇒ d B d t = λ B ,   λ   is proportionility constant ⇒ d B B = λ d t Integrate both sides with proper limit we get, ∫ B 1 B 2 d B B = ∫ t 1 t 2 λ d t ⇒ ln B 2 B 1 = λ t 2 - t 1     . . . i Given that, at t = 0 ,   B = 1000   &   t = 20 %   increase = 1200 ∴   ln 1200 1000 = 2 λ ⇒ λ = 1 2 ln 6 5 Now an equation i   becomes, ln B 2 B 1 = 1 2 ln 6 5 t 2 - t