Given:
Initial number of bacteria = 3000
With a growth constant (k) of 2.8 per hour.
To find:
The number of hours it will take to be 15,000 bacteria.
Solution:
Let P(t) be the number of bacteria after t number of hours.
The exponential growth model (continuously) is:
[tex]P(t)=P_0e^{kt}[/tex]
Where, [tex]P_0[/tex] is the initial value, k is the growth constant and t is the number of years.
Putting [tex]P(t)=15000,P_0=3000, k=2.8[/tex] in the above formula, we get
[tex]15000=3000e^{2.8t}[/tex]
[tex]\dfrac{15000}{3000}=e^{2.8t}[/tex]
[tex]5=e^{2.8t}[/tex]
Taking ln on both sides, we get
[tex]\ln 5=\ln e^{2.8t}[/tex]
[tex]1.609438=2.8t[/tex] [tex][\because \ln e^x=x][/tex]
[tex]\dfrac{1.609438}{2.8}=t[/tex]
[tex]0.574799=t[/tex]
[tex]t\approx 0.575[/tex]
Therefore, the number of bacteria will be 15,000 after 0.575 hours.
