Answer:
The size of the colony after 4 days is 8351.15.
8 days long there are 70,000 mosquitoes.
Step-by-step explanation:
Given : The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1700 after 1 day.
To find : What is the size of the colony after 4 days and How long is it until there are 70,000 mosquitoes?
Solution :
Let the uninhibited growth is defined by a function,
![A=A_0e^{kt}](https://tex.z-dn.net/?f=A%3DA_0e%5E%7Bkt%7D)
Where,
is the initial amount
e is the Euler's constant
k is the amount of increase
t=1 day is the time
A=1700 is the amount
Substitute all the values in the formula,
![1700=1000e^{k(1)}](https://tex.z-dn.net/?f=1700%3D1000e%5E%7Bk%281%29%7D)
![\frac{1700}{1000}=e^{k(1)}](https://tex.z-dn.net/?f=%5Cfrac%7B1700%7D%7B1000%7D%3De%5E%7Bk%281%29%7D)
![1.7=e^{k}](https://tex.z-dn.net/?f=1.7%3De%5E%7Bk%7D)
Taking natural log both side,
![\ln(1.7)=\ln(e^{k})](https://tex.z-dn.net/?f=%5Cln%281.7%29%3D%5Cln%28e%5E%7Bk%7D%29)
![0.5306=k](https://tex.z-dn.net/?f=0.5306%3Dk)
Now, The size of the colony after 4 days is
![A=1000e^{(0.5306)(4)}](https://tex.z-dn.net/?f=A%3D1000e%5E%7B%280.5306%29%284%29%7D)
![A=1000e^{2.1224}](https://tex.z-dn.net/?f=A%3D1000e%5E%7B2.1224%7D)
![A=1000\times 8.35115](https://tex.z-dn.net/?f=A%3D1000%5Ctimes%208.35115)
![A=8351.15](https://tex.z-dn.net/?f=A%3D8351.15)
Therefore, The size of the colony after 4 days is 8351.15.
When there are 70,000 mosquitoes the time is
![70000=1000e^{(0.5306)(t)}](https://tex.z-dn.net/?f=70000%3D1000e%5E%7B%280.5306%29%28t%29%7D)
![\frac{70000}{1000}=e^{(0.5306)(t)}](https://tex.z-dn.net/?f=%5Cfrac%7B70000%7D%7B1000%7D%3De%5E%7B%280.5306%29%28t%29%7D)
![70=e^{(0.5306)(t)}](https://tex.z-dn.net/?f=70%3De%5E%7B%280.5306%29%28t%29%7D)
Taking ln both side,
![\ln(70)=\ln(e^{0.5306t})](https://tex.z-dn.net/?f=%5Cln%2870%29%3D%5Cln%28e%5E%7B0.5306t%7D%29)
![4.248=0.5306t](https://tex.z-dn.net/?f=4.248%3D0.5306t)
![t=\frac{4.248}{0.5306}](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B4.248%7D%7B0.5306%7D)
![t=8](https://tex.z-dn.net/?f=t%3D8)
Therefore, 8 days long there are 70,000 mosquitoes.