Answer:
a) 13032
b)6.4 days
Step-by-step explanation:
The formula for the population growth is given by
[tex]A=Pe^{rt}[/tex]
Here, A = final population, P = initial population, r = growth rate and t = time
From the given directions, we have
A = 1900, P = 1000, t = 1, r = ?
Substituting these values in the above formula to find r
[tex]1900=1000e^{r\cdot1}\\\\1900=1000e^r[/tex]
Divide both sides by 1000
[tex]\frac{1900}{1000}=e^r\\\\e^r=\frac{19}{10}[/tex]
Take natural log both sides
[tex]\ln(e^r)=\ln(\frac{19}{10})\\\\r=0.64185[/tex]
Therefore, the population model is given by
[tex]A=1000e^{0.64185t}[/tex]
(a) The size of the colony after 4 days is given by
[tex]A=1000e^{0.64185\cdot4}\\\\A=13032[/tex]
(B) The time for the number of mosquitoes to be 60,000 is
[tex]60000=1000e^{0.64185t}[/tex]
Divide both sides by 1000
[tex]\frac{60000}{1000}=e^{0.64185t}\\\\e^{0.64185t}=60[/tex]
Take natural log both sides
[tex]\ln(e^{0.64185t})=\ln60\\\\0.64185t=\ln60\\\\t=\frac{\ln60}{0.64185}\\\\t=6.4[/tex]
Hence, it will take 6.4 days to the population of mosquitoes to be 60000.
