Respuesta :
Exponential Continuous Growth
The exponential is commonly used to model natural processes of growth.
The formula of the exponential continuous growth is:
[tex]P=P_o\cdot e^{kt}[/tex]Where:
Po = the initial population of the bacteria culture
k = a fixed constant
t = time
P = the population of the bacteria culture at any time t
The initial population of bacteria is given as Po=10
We are given the growth rate at 1.5% per minute.
The process starts at 7:00 am. At 7:01 am there will be 1.5% more bacteria than the previous minute, that is P=1.015*10=10.15 for t=1, thus:
[tex]10.15=10\cdot e^{k(1)}=10\cdot e^k[/tex]Solving for k:
[tex]\begin{gathered} 10\cdot e^k=10.15 \\ e^k=1.015 \\ k=\ln \text{ 1.015} \\ k=0.01489 \end{gathered}[/tex]Now we have the value of k, the function is:
[tex]P(t)=10\cdot e^{0.01489t}[/tex]We are required to find the bacteria count after 12 hours. Since the time must be expressed in minutes, t=12 hours = 12*60 = 720 minutes
[tex]P=10\cdot e^{0.01489\cdot720}=10\cdot45,242.9=452,429[/tex]The bacteria count will be 452,429 after 12 hours
