Answer:
The number of bacteria after t hours is given by: [tex]y(t) = 100e^{0.3219t}[/tex]
Step-by-step explanation:
Amount of bacteria after t hours:
Is given by the following equation:
[tex]y(t) = ae^{kt}[/tex]
In which a is the initial value and k is the constant of growth.
There are 100 bacteria initially
This means that [tex]a = 100[/tex]. So
[tex]y(t) = ae^{kt}[/tex]
[tex]y(t) = 100e^{kt}[/tex]
500 bacteria five hours later.
This means that [tex]y(5) = 500[/tex]. We use this to find k. So
[tex]y(t) = 100e^{kt}[/tex]
[tex]500 = 100e^{5k}[/tex]
[tex]e^{5k} = 5[/tex]
[tex]\ln{e^{5k}} = \ln{5}[/tex]
[tex]5k = \ln{5}[/tex]
[tex]k = \frac{\ln{5}}{5}[/tex]
[tex]k = 0.3219[/tex]
So
[tex]y(t) = 100e^{kt}[/tex]
[tex]y(t) = 100e^{0.3219t}[/tex]
The number of bacteria after t hours is given by: [tex]y(t) = 100e^{0.3219t}[/tex]
