Answer :
Answer:
The hourly growth rate parameter of the bacteria is 3%.
Step-by-step explanation:
Continuous exponential growth formula:
[tex]A=Pe^{rt}[/tex]
A= Population of bacteria after t hours
P = Initial population
r= growth rate
t= time in hours
Given that,
A sample of 2200 bacteria selected from the population reached the size 2300 bacteria in one and half hours.
A=2300, P=2200, t [tex]=1\frac12[/tex] hours =[tex]\frac32[/tex] hours,r=?
[tex]\therefore2300=2200e^{\frac32r}[/tex]
[tex]\Rightarrow2200e^{\frac32r}=2300[/tex]
[tex]\Rightarrow e^{\frac32r}=\frac{2300}{2200}[/tex]
[tex]\Rightarrow e^{\frac32t}=\frac{23}{22}[/tex]
Taking ln both sides
[tex]\Rightarrow ln( e^{\frac32r})=ln(\frac{23}{22})[/tex]
[tex]\Rightarrow {\frac32r}=ln(\frac{23}{22})[/tex]
[tex]\Rightarrow r}=\frac{ln|\frac{23}{22}|}{\frac32}[/tex]
[tex]\Rightarrow r}\approx0.030[/tex]
[tex]\Rightarrow r}=3\%[/tex]
The hourly growth rate parameter of the bacteria is 3%.
The hourly growth rate is 2%
An exponential growth is in the form:
y = abˣ;
where y, x are variables, a is the initial value of y and b > 1
Let y represent the number of bacteria after x hours.
There was initially 2200 bacteria, hence a = 2200. The equation becomes:
[tex]y=2200(b)^x\\\\\\After\ 1.5\ hours,y=2300:\\\\\\2300=2200(b)^{1.5}\\\\\\b^{1.5}=1.045\\\\1.5lnb=ln(1.045)\\\\ln(b)=0.0296\\\\b=1.02[/tex]
Therefore the hourly growth rate is 1.02 - 1 = 0.02 = 2%
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