# An initial population of 505 quail increases at an annual rate of 23%. How do you write an exponential function to model the quail population?

May 19, 2017

$y = 505 {\left(1 + 0.23\right)}^{t}$

#### Explanation:

Population growth is seen to increase at an exponential rate. We use the following to model this growth.

$\textcolor{w h i t e}{a a a a a a a a a a a a a} y = {A}_{o} {\left(1 + r\right)}^{t}$

Where
$\text{y = value at time (t)}$
$\text{A"_(o) = "original value}$
$\text{r = rate of growth}$
$\text{t = time elapsed}$

An exponential function that models the quail population is set up like...

color(white)(aaaaaaaaaaaaa)color(orange)(y = 505(1+0.23)^(t)

so that, for example, if we wanted to figure out the population at time (3 years), then we would set up the function as follows,

$y = 505 {\left(1 + 0.23\right)}^{3}$

to get

$y = 939.3 \approx 939 \text{ quails}$ after the ${3}^{r d}$ year has elapsed.