Question

In ideal conditions, 200 colony-forming units (cfu) of E. coli can grow to 400 cfu in 20 minutes, to 800 cfu in 40 minutes, and to 1600 cfu in an hour.

Part A: Jenny says that the E. coli growth can be modeled with an exponential function. Do you agree? Justify your answer.

Part B: Write the equation that models the function.Immersive Reader

Answer 1

(a)Yes, it is an exponential function.

(b)
`P(t)=200\cdot 2^(t/20)`

Explanation:

(a)A population grows exponentially if it increases by a common ratio(called the growth ratio). We can see from the given information that the population of E.coli doubles every 20 minutes, therefore it is an exponential growth with a growth ratio of 2.

(b)The population, P(t) at any time t of an initial population,
`P_o` with a growth ratio of r over a period k can be modeled using the function:

`P(t)=P_o\cdot r^(t/k)`

In this Case:

`P_o`=200

Growth rate,r=2

Growth Period,k=20 minutes

Therefore, the equation that models the function is:

`P(t)=200\cdot 2^(t/20)` , t is in minutes