Question

Ryan spills some soda and neglects to clean it up. When leaving for spring break, Ryan notices some ants on the sticky mess, but ignores them. Upon returning 17 days later, Ryan counts 3960 ants in the same place. The next day there are 5280 ants. Assuming that the size of the ant population can be described exponentially, find the number of ants that Ryan saw when he left for spring break

Answer 1

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:

$P(t) = P(0)e^{rt}$

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 $P(17) = 3960$. So

$P(t) = P(0)e^{rt}$

$3960 = P(0)e^{17r}$

The next day there are 5280 ants.

This means that $P(18) = 5280$

So

$P(t) = P(0)e^{rt}$

$5280 = P(0)e^{18r}$

From above, we have that

$P(0) = \frac{3960}{e^{17r}}$

Replacing

$5280 = P(0)e^{18r}$

$5280 = \frac{3960e^{18r}}{e^{17r}}$

$3960e^{r} = 5280$

$e^{r} = \frac{5280}{3960}$

$\ln{e^{r}} = \ln{\frac{5280}{3960}}$

$r = \ln{\frac{5280}{3960}}$

$r = 0.2877$

Find the number of ants that Ryan saw when he left for spring break

$P(0) = \frac{3960}{e^{17r}}$

$P(0) = \frac{3960}{e^{17*0.2877}}$

$P(0) = 30$

Ryan saw 30 ants when he left for spring break.