Respuesta :
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:
[tex]P(t) = P(0)e^{rt}[/tex]
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 [tex]P(0) = 80, P(4) = 145[/tex]
We use this to find the growth rate r for the population of termites.
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]145 = 80e^{4r}[/tex]
[tex]e^{4r} = \frac{145}{80}[/tex]
[tex]\ln{e^{4r}} = \ln{\frac{145}{80}}[/tex]
[tex]4r = \ln{\frac{145}{80}}[/tex]
[tex]r = \frac{\ln{\frac{145}{80}}}{4}[/tex]
[tex]r = 0.1487[/tex]
The termites population is modeled by:
[tex]P(t) = 80e^{0.1487t}[/tex]
Spiders:
Three days after moving in, there are two times as many termites as spiders.
[tex]P(3) = 80e^{0.1487*3} = 125[/tex]
There are 125 termites, so there are 62 spiders.
Eight days after moving in, there were four times as many termites as spiders.
[tex]P(8) = 80e^{0.1487*8} = 263[/tex]
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
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]62 = P(0)e^{3r}[/tex]
And
[tex]65 = P(0)e^{8r}[/tex]
From the first equation:
[tex]P(0) = \frac{62}{e^{3r}}[/tex]
Replacing in the second:
[tex]65 = \frac{62e^{8r}}{e^{3r}}[/tex]
[tex]e^{5r} = \frac{65}{62}[/tex]
[tex]\ln{e^{5r}} = \ln{\frac{65}{62}}[/tex]
[tex]5r = \ln{\frac{65}{62}}[/tex]
[tex]r = \frac{\ln{\frac{65}{62}}}{5}[/tex]
[tex]r = 0.0095[/tex]
So
[tex]P(t) = P(0)e^{0.0095t}[/tex]
How long (in days) does it take the population of spiders to triple
This is t for which P(t) = 3P(0). So
[tex]P(t) = P(0)e^{0.0095t}[/tex]
[tex]3P(0) = P(0)e^{0.0095t}[/tex]
[tex]e^{0.0095t} = 3[/tex]
[tex]\ln{e^{0.0095t}} = \ln{3}[/tex]
[tex]0.0095t = \ln{3}[/tex]
[tex]t = \frac{\ln{3}}{0.0095}[/tex]
[tex]t = 115.64[/tex]
Rounding to the nearest whole number
It takes 116 days for the population of spiders to triple
