Answer:
It takes 116 days for the population of spiders to triple
Step-by-step explanation:
The exponential model for population growth is given by:
![P(t) = P(0)e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7Brt%7D)
In which P(t) is the population after t days, P(0) is the initial population and r is the growth rate.
Termites:
The house contains 80 termites the day you move in. After four days, the house contains 145 termites.
This means that ![P(0) = 80, P(4) = 145](https://tex.z-dn.net/?f=P%280%29%20%3D%2080%2C%20P%284%29%20%3D%20145)
We use this to find the growth rate r for the population of termites.
![P(t) = P(0)e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7Brt%7D)
![145 = 80e^{4r}](https://tex.z-dn.net/?f=145%20%3D%2080e%5E%7B4r%7D)
![e^{4r} = \frac{145}{80}](https://tex.z-dn.net/?f=e%5E%7B4r%7D%20%3D%20%5Cfrac%7B145%7D%7B80%7D)
![\ln{e^{4r}} = \ln{\frac{145}{80}}](https://tex.z-dn.net/?f=%5Cln%7Be%5E%7B4r%7D%7D%20%3D%20%5Cln%7B%5Cfrac%7B145%7D%7B80%7D%7D)
![4r = \ln{\frac{145}{80}}](https://tex.z-dn.net/?f=4r%20%3D%20%5Cln%7B%5Cfrac%7B145%7D%7B80%7D%7D)
![r = \frac{\ln{\frac{145}{80}}}{4}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B%5Cln%7B%5Cfrac%7B145%7D%7B80%7D%7D%7D%7B4%7D)
![r = 0.1487](https://tex.z-dn.net/?f=r%20%3D%200.1487)
The termites population is modeled by:
![P(t) = 80e^{0.1487t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%2080e%5E%7B0.1487t%7D)
Spiders:
Three days after moving in, there are two times as many termites as spiders.
![P(3) = 80e^{0.1487*3} = 125](https://tex.z-dn.net/?f=P%283%29%20%3D%2080e%5E%7B0.1487%2A3%7D%20%3D%20125)
There are 125 termites, so there are 62 spiders.
Eight days after moving in, there were four times as many termites as spiders.
![P(8) = 80e^{0.1487*8} = 263](https://tex.z-dn.net/?f=P%288%29%20%3D%2080e%5E%7B0.1487%2A8%7D%20%3D%20263)
There are 263 termites, so there are 65 spiders(i am rounding down the number of spiders).
Building the system for spiders:
P(3) = 62 and P(8) = 65
Then
![P(t) = P(0)e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7Brt%7D)
![62 = P(0)e^{3r}](https://tex.z-dn.net/?f=62%20%3D%20P%280%29e%5E%7B3r%7D)
And
![65 = P(0)e^{8r}](https://tex.z-dn.net/?f=65%20%3D%20P%280%29e%5E%7B8r%7D)
From the first equation:
![P(0) = \frac{62}{e^{3r}}](https://tex.z-dn.net/?f=P%280%29%20%3D%20%5Cfrac%7B62%7D%7Be%5E%7B3r%7D%7D)
Replacing in the second:
![65 = \frac{62e^{8r}}{e^{3r}}](https://tex.z-dn.net/?f=65%20%3D%20%5Cfrac%7B62e%5E%7B8r%7D%7D%7Be%5E%7B3r%7D%7D)
![e^{5r} = \frac{65}{62}](https://tex.z-dn.net/?f=e%5E%7B5r%7D%20%3D%20%5Cfrac%7B65%7D%7B62%7D)
![\ln{e^{5r}} = \ln{\frac{65}{62}}](https://tex.z-dn.net/?f=%5Cln%7Be%5E%7B5r%7D%7D%20%3D%20%5Cln%7B%5Cfrac%7B65%7D%7B62%7D%7D)
![5r = \ln{\frac{65}{62}}](https://tex.z-dn.net/?f=5r%20%3D%20%5Cln%7B%5Cfrac%7B65%7D%7B62%7D%7D)
![r = \frac{\ln{\frac{65}{62}}}{5}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B%5Cln%7B%5Cfrac%7B65%7D%7B62%7D%7D%7D%7B5%7D)
![r = 0.0095](https://tex.z-dn.net/?f=r%20%3D%200.0095)
So
![P(t) = P(0)e^{0.0095t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7B0.0095t%7D)
How long (in days) does it take the population of spiders to triple
This is t for which P(t) = 3P(0). So
![P(t) = P(0)e^{0.0095t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7B0.0095t%7D)
![3P(0) = P(0)e^{0.0095t}](https://tex.z-dn.net/?f=3P%280%29%20%3D%20P%280%29e%5E%7B0.0095t%7D)
![e^{0.0095t} = 3](https://tex.z-dn.net/?f=e%5E%7B0.0095t%7D%20%3D%203)
![\ln{e^{0.0095t}} = \ln{3}](https://tex.z-dn.net/?f=%5Cln%7Be%5E%7B0.0095t%7D%7D%20%3D%20%5Cln%7B3%7D)
![0.0095t = \ln{3}](https://tex.z-dn.net/?f=0.0095t%20%3D%20%5Cln%7B3%7D)
![t = \frac{\ln{3}}{0.0095}](https://tex.z-dn.net/?f=t%20%3D%20%5Cfrac%7B%5Cln%7B3%7D%7D%7B0.0095%7D)
![t = 115.64](https://tex.z-dn.net/?f=t%20%3D%20115.64)
Rounding to the nearest whole number
It takes 116 days for the population of spiders to triple