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# The rate of increase of bacteria in a culture is proportional to the number of bacteria present, and it is found that the number doubles in 6 hours. Prove that the bacteria becomes eight times at the end of 18 hours.

Let the count of bacteria be C at any time t.

According to the question,

where k is a constant

Integrating both sides, we have

= k∫dt

log|C| = kt + a……(1)

Given, we have C = C0 when t = 0 sec

Putting the value in equation (1)

log|C| = kt + a

log| C0| = 0 + a

a = log| C0| ……(2)

Putting the value of a in equation (1) we have,

log|C| = kt + log| C0|

log|C| – log| C0| = k t []

log ( = kt ……(3)

Also, at t = 6 years, C = 2C0

From equation(3),we have

kt = log (

k×6 = log (

k = ……(4)

Now, equation (3) becomes,

Now, C = 8C0

t = 18

The time required = 18 hours

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