Question

A culture started with 2,000 bacteria. After 4 hours it grew to 2,600 bacteria. Predict how many bacteria will be present after 17 hours. Round your answer to the nearest whole number.

Answer 1

1. The population of bacteria after time t, is given by the formula:

P(t)=
`I e^(kt)`,

where I is the initial population and k the growth constant.

For example, the population at time t=0 is P(0)=
`I e^(k0)=Ie^(0)=I*1=I`

2.
P(t)=
`2000 e^(kt)`

we can use the information we have:

P(4)=2,600=
`2000 e^(4k)`


`( e^(k) )^(4)= (2,600)/(2000)`


`( e^(k) )^(4)= 1.3`


`e^(k)= \sqrt[4]{1.3} = \sqrt[2]{ \sqrt[2]{1.3} }= \sqrt[2]{1.14} =1.068`

3.
So the function becomes P(t)=
`2000 e^(kt)=2000 ( e^(k)) {t} =2000 (1.068)^(t)`

4. Population after 17 hours is P(17)=
`2000(1.068)^(17)=2000*3.06=` 6120