Question

The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that 500 bacteria are present. After 10 hours 5000 bacteria are present. What was the initial number of bacteria

Answer 1

The number of bacteria at initial = 187

Explanation:

Given that the population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t.

`(dN)/(dt) = k N`

`(dN)/(N) = k dt`

Integrating both side we get

㏑ N = k t + C ------- (1)

Now given that after 3 hours it is observed that 500 bacteria are present and after 10 hours 5000 bacteria are present.

⇒ ㏑ 500 = 3 k + C -------- (2)

⇒ ㏑ 5000 = 10 k + C ------ (3)

⇒ ㏑ 5000 - ㏑ 500 = 7 k

⇒ ㏑
`(5000)/(500)` = 7 k

⇒ ㏑ 10 = 7 k

⇒ k = 0.329

Put this value of k in equation (2),

⇒ ㏑ 500 = 3 × 0.329 + C

⇒ C = 5.23

Put this value of C in equation 1 we get,

⇒ ㏑ N = k t + 5.23

Initially when t = 0 , then

⇒ ㏑ N = 5.23

⇒ N = 187

Thus the number of bacteria at initial = 187