To model the population growth of the quail, we'll use an exponential function that incorporates the initial population, the growth rate, and the time in years.
### Step-by-Step Solution:
1. Define the Initial Population:
The initial population of the quail is [tex]\( P_0 = 700 \)[/tex].
2. Identify the Growth Rate:
The annual growth rate is 30%. In decimal form, this is [tex]\( r = 0.30 \)[/tex].
3. Determine the General Form of the Exponential Function:
An exponential growth function can be written as:
[tex]\[
P(t) = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time in years.
4. Substitute the Given Values:
Substitute [tex]\( P_0 = 700 \)[/tex] and [tex]\( r = 0.30 \)[/tex] into the general formula:
[tex]\[
P(t) = 700 \times (1 + 0.30)^t
\][/tex]
5. Simplify the Expression:
Simplify the term inside the parentheses:
[tex]\[
P(t) = 700 \times 1.30^t
\][/tex]
So, the exponential function to model the quail population can be written as:
[tex]\[
P(t) = 700 \times 1.30^t
\][/tex]
This function [tex]\( P(t) = 700 \times 1.30^t \)[/tex] depicts how the population of quails grows exponentially over time, with 700 being the starting population and 1.30 representing the 30% annual increase.
