To estimate the population of an endangered species over time, we need to take into account the initial population and the rate at which the population is decreasing annually. Here is how we approach the solution step-by-step:
1. Initial Population:
The initial population of the endangered species is given as 825.
2. Annual Decline Rate:
The population decreases by 7.5% each year. To use this in calculations, we need to convert this percentage into a decimal:
[tex]\[
\text{Decline rate} = \frac{7.5}{100} = 0.075
\][/tex]
3. Retention Rate:
Instead of focusing on the decline, let's consider the retention rate, which is the proportion of the population that remains after each year. The retention rate is calculated as:
[tex]\[
\text{Retention rate} = 1 - \text{Decline rate} = 1 - 0.075 = 0.925
\][/tex]
4. Population Estimation Over Time:
If [tex]\( x \)[/tex] represents the number of years, the population at any given year can be estimated using the initial population and the retention rate raised to the power of [tex]\( x \)[/tex]:
[tex]\[
P = 825 \times (0.925)^x
\][/tex]
Where [tex]\( P \)[/tex] is the population after [tex]\( x \)[/tex] years, 825 is the initial population, and 0.925 is the annual retention rate.
Based on the given options, the correct rule to estimate the population of the endangered animal over time is:
[tex]\[
P = 825 \times (0.925)^x
\][/tex]
Thus, the correct answer is:
[tex]\[
P = 825(0.925)^x
\][/tex]
