He 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 400 bacteria are present. After 10 hours 4000 bacteria are present. What was the initial number of bacteria

Explanation:

P(t) =

Population after 3 hours is 400

i.e., P(3) = 400

400 =                                   .................(1)

Population after 10 hours is 2000 i.e. P(5) = 2000

2000 =                               .................(2)

From eq. (1) we get

From eq. (2) we get

Compare above two equations and solve for k

By taking log on both sides, we get

- 3k =

⇒ - 3k =

⇒ - 3k =

⇒ 7k =

⇒ k = = 0.2299

Put k = 0.2299 in equation (1), we get

400 =  

⇒ 400 =  

⇒

 ≈ 201 bacteria