Question

bacteria culture starts with 400 bacteria and grows at a rate proportional to its size. After 4 hours, there are 9000 bacteria. A. Find an expression for the number of bacteria after tt hours. B. Find the number of bacteria after 5 hours. C. After how many hours will the population reach 30000

Answer 1

A)
`P = P_0 e^(rt)`

B)
`19564`

C)
`5.5` or
`6` hours

Step-by-step explanation:

A) The formula that can be used for bacteria after "t" hours is

`P = P_0 e^(rt)`

Where ,

`P` represents the population after "t" hours

`P_0` represents the initial population.

`r` represents the rate of growth of population

`t` represents the time taken by the population to grow

B) Here let us first find the rate of population growth.

`9000 = 400 * e^(r*4)\\3.11 = r* 4\\r = 0.778`

Thus, the population of bacteria after 5 hours

`P = 400 * e^(0.778 * 5) \\P = 400* 48.91\\P =  19564\\`

C)

`30000 = 400 * e^{0.778 * t) \\\\4.317 = 0.778 * t\\t = 5.549\\or t= 5.5 or 6`