Given that a species of beetles grows 32% every year.
So growth rate is given by
r=32%= 0.32
Given that 100 beetles are released into a field.
So that means initial number of beetles P=100
Now we have to find about how many beetles will there be in 10 years.
To find that we need to setup growth formula which is given by
where A is number of beetles at any year n.
Plug the given values into above formula we get:
![A=100(1+0.32)^n](https://tex.z-dn.net/?f=A%3D100%281%2B0.32%29%5En)
![A=100(1.32)^n](https://tex.z-dn.net/?f=A%3D100%281.32%29%5En)
now plug n=10 years
![A=100(1.32)^{10}=100(16.0597696605)=1605.97696605](https://tex.z-dn.net/?f=A%3D100%281.32%29%5E%7B10%7D%3D100%2816.0597696605%29%3D1605.97696605)
Hence answer is approx 1606 beetles will be there after 10 years.
To find answer for 20 years plug n=20 years
![A=100(1.32)^{20}=100(257.916201549)=25791.6201549](https://tex.z-dn.net/?f=A%3D100%281.32%29%5E%7B20%7D%3D100%28257.916201549%29%3D25791.6201549)
Hence answer is approx 25791 beetles will be there after 20 years.
Now we have to find time for 100000 beetles so plug A=100000
![A=100(1.32)^n](https://tex.z-dn.net/?f=A%3D100%281.32%29%5En)
![100000=100(1.32)^n](https://tex.z-dn.net/?f=100000%3D100%281.32%29%5En)
![1000=(1.32)^n](https://tex.z-dn.net/?f=1000%3D%281.32%29%5En)
![log(1000)=n*log(1.32)](https://tex.z-dn.net/?f=log%281000%29%3Dn%2Alog%281.32%29)
24.8810001465=n
Hence answer is approx 25 years.