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

Step-by-step solution

Let ' x ' be the number of bacteria present at time ' t '. d x dt x d x dt = x , where is the constant of proportionality Integrating on both sides, we get x= t+c When t =0, x=1000 array ll & 1000=0+c & c = (1000) & x= t + (1000)...(i) & When t =2 & x=1000+(20 % of 1000) & =1000+200 & =1200 array array ll & 1200=2 + 1000 & = 1 2 ( 1200 1000 )= 1 2 ( 6 5 ) & x= t 2 ( 6 5 )+ (1000) array ...[From (i)] aligned & When t = k ( 6 5 ) , x=2000 & 2000= k ( 6 5 ) 1 2 ( 6 5 )+ (1000) & ( 2000 1000 )= k 2 2= k 2 & & k 2 =2 &