Let y denote the number of bacteria at any instant t . then according to the question dy‌dt‌α‌y⇒
dy
y
=k‌dt ... (i) k is the constant of proportionality, taken to be + ve on integrating (i), we get log‌y=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=log‌y0 ⇒ log‌y=kt+log‌y0 ⇒log‌
y
y0
=kt ... (iii) y=(y0+
10
100
y0)=
11y0
10
, when t=2, So, from (iii), we get log‌
11y0
10
y0
=k (2) ⇒ k=
1
2
‌log‌
11
10
...(iv) Using (iv) in (iii) log‌
y
y0
=
1
2
(log‌
11
10
)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