Respuesta :
Answer:
There will be 50 bacteria remaining after 28 minutes.
Step-by-step explanation:
The exponential decay equation is
[tex]N=N_0e^{-rt}[/tex]
N= Number of bacteria after t minutes.
[tex]N_0[/tex] = Initial number of bacteria when t=0.
r= Rate of decay per minute
t= time is in minute.
The sample begins with 500 bacteria and after 11 minutes there are 200 bacteria.
N=200
[tex]N_0[/tex] = 500
t=11 minutes
r=?
[tex]N=N_0e^{-rt}[/tex]
[tex]\therefore 200=500e^{-11r}[/tex]
[tex]\Rightarrow e^{-11r}=\frac{200}{500}[/tex]
Taking ln both sides
[tex]\Rightarrow ln| e^{-11r}|=ln|\frac{2}{5}|[/tex]
[tex]\Rightarrow {-11r}=ln|\frac{2}{5}|[/tex]
[tex]\Rightarrow r}=\frac{ln|\frac{2}{5}|}{-11}[/tex]
To find the time when there will be 50 bacteria remaining, we plug N=50, [tex]N_0[/tex]= 500 and  [tex]r}=\frac{ln|\frac{2}{5}|}{-11}[/tex] in exponential decay equation.
[tex]50=500e^{-\frac{ln|\frac25|}{-11}.t}[/tex]
[tex]\Rightarrow \frac{50}{500}=e^{\frac{ln|\frac25|}{11}.t}[/tex]
Taking ln both sides
[tex]\Rightarrow ln|\frac{50}{500}|=ln|e^{\frac{ln|\frac25|}{11}.t}|[/tex]
[tex]\Rightarrow ln|\frac{1}{10}|={\frac{ln|\frac25|}{11}.t}[/tex]
[tex]\Rightarrow t= \frac{ln|\frac{1}{10}|}{\frac{ln|\frac25|}{11}.}[/tex]
[tex]\Rightarrow t= \frac{11\times ln|\frac{1}{10}|}{{ln|\frac25|}}[/tex]
[tex]\Rightarrow t\approx 28[/tex] minutes
There will be 50 bacteria remaining after 28 minutes.
