Question

at the beginning of a story, a certain culture of bacteria has a population of 80. the population grows according to a continuous exponential growth model. after 14 days, there are 216 bacteria.

Answer 1

Given:

The population of the bacteria at the beginning = 80

the population grows according to a continuous exponential growth model.

after 14 days, there are 216 bacteria. ​

y = the number of bacteria after time t

So, the general relation between y and t will be:

`y=a\cdot e^(bt)`

We need to find the values of a and b

At t = 0 y = 80

So,

`\begin{gathered} 80=a\cdot e^0 \\ a=80 \end{gathered}`

When t = 14 , y = 216

So,

`\begin{gathered} 216=80e^(14b) \\ (216)/(80)=e^(14b) \\ \text{2}.7=e^(14b) \\ \ln 2.7=14b \\ \text{0}.99325=14b \\ b=(0.99325)/(14)=0.071 \end{gathered}`

so, the function will be:

`y=80\cdot e^(0.071t)`

Part b: we need to find the number of bacteria after 23 days

So, substitute with t = 23

so,

`y=80\cdot e^(0.071\cdot23)=409`

so, after 23 days the number of bacteria = 409