Question

A bacteria culture starts with 200 bacteria and grows at a rate proportional to its size. After 6 hours there will be 1200 bacteria (1) Express the population after I hours as a function of t. population: p(tepe (1.066-21) (unction of t) (b) What will be the population after 7 hours? 348125.2 (c) How long will it take for the population to reach 1750 ? Note: You can earn partial credit on this problem.

Answer 1

We are given that the rate of change is proportional to its size S

So,
`(dS)/(dt) \propto S`

`(dS)/(dt) = kS`

`(dS)/(S) = kdt`

Integrating both sides

`\log(S)= kt + log c`

`(S)/(S_0)=e^(kt)`

`S=S_0 e^(kt)`

S is the population after t hours

`S_0` is the initial population

Now we are given that After 6 hours there will be 1200 bacteria

`1200=200 e^(6k)`

`6=e^(6k)`

`6^{(1)/(6)=e^(k)`

So,
`S=200 * 6^{(t)/(6)`

a)Now the population after t hours as a function of t;
`S=200 * 6^{(t)/(6)`

b) What will be the population after 7 hours?

Substitute t = 7 hours

A bacteria culture starts with 200 bacteria

`S=200 * 6^{(7)/(6)}`

`S=1617.607`

c) How long will it take for the population to reach 1750 ?

`1750=200 * 6^{(t)/(6)`

`(1750)/(200) =6^{(t)/(6)`

`8.75 =6^{(t)/(6)`

`t=7.26`

So, it will take 7.2 hours for the population to reach 1750