Let y denote the number of bacteria at any instant t • then according to the question
αy ⇒
= kdt ... (i)
k is the constant of proportionality, taken to be + ve on integrating (i), we get
log y = kt + c ...(ii)
c is a parameter, let
y0 be the initial number of bacteria
i.e., at t = 0 using this in (ii), c = log
y0 ⇒ log y = kt + log
y0 ⇒ log
= kt ... (iii)
y =
(y0+y) =
, when t = 2
So, from (iii), we get log
= k (2)
⇒ k =
log
... (iv)
Using (iv) in (iii) log
=
(log) t ... (v)
let the number of bacteria become 1, 00, 000 to 2,00,000 in
t1 hours, i.e., y =
2y0 when t =
t1 hours, from (v)
log
=
(log)t1 ⇒
t1 =
Hence, the reqd. no. of hours =