Respuesta :
Answer:
Ryan saw 30 ants when he left for spring break.
Step-by-step explanation:
The equation for the number of ants has the following format:
[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.
Upon returning 17 days later, Ryan counts 3960
This means that [tex]P(17) = 3960[/tex]. So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]3960 = P(0)e^{17r}[/tex]
The next day there are 5280 ants.
This means that [tex]P(18) = 5280[/tex]
So
[tex]P(t) = P(0)e^{rt}[/tex]
[tex]5280 = P(0)e^{18r}[/tex]
From above, we have that
[tex]P(0) = \frac{3960}{e^{17r}}[/tex]
Replacing
[tex]5280 = P(0)e^{18r}[/tex]
[tex]5280 = \frac{3960e^{18r}}{e^{17r}}[/tex]
[tex]3960e^{r} = 5280[/tex]
[tex]e^{r} = \frac{5280}{3960}[/tex]
[tex]\ln{e^{r}} = \ln{\frac{5280}{3960}}[/tex]
[tex]r = \ln{\frac{5280}{3960}}[/tex]
[tex]r = 0.2877[/tex]
Find the number of ants that Ryan saw when he left for spring break
[tex]P(0) = \frac{3960}{e^{17r}}[/tex]
[tex]P(0) = \frac{3960}{e^{17*0.2877}}[/tex]
[tex]P(0) = 30[/tex]
Ryan saw 30 ants when he left for spring break.
