Question
The elephant population of a nature preserve since 1990 is modeled by the exponential function f(x)=315⋅1.075x . Find the elephant population in 1990 and the percentage at which the population increases each year.
Answers
GPT-4o mini
To find the elephant population in 1990 using the given model \( f(x) = 315 \cdot 1.075^x \), we need to evaluate the function at \( x = 0 \), since \( x \) represents the number of years since 1990.
1. **Finding the population in 1990:**
\[
f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315
\]
Therefore, the elephant population in 1990 was **315**.
2. **Finding the percentage increase in population each year:**
The term \( 1.075 \) in the exponential function represents the growth factor. To find the percentage increase, we can calculate:
\[
\text{Percentage Increase} = (1.075 - 1) \times 100\% = 0.075 \times 100\% = 7.5\%
\]
Thus, the percentage increase in the elephant population each year is **7.5%**.
In summary:
- The elephant population in 1990 was **315**.
- The population increases by **7.5%** each year.
1. **Finding the population in 1990:**
\[
f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315
\]
Therefore, the elephant population in 1990 was **315**.
2. **Finding the percentage increase in population each year:**
The term \( 1.075 \) in the exponential function represents the growth factor. To find the percentage increase, we can calculate:
\[
\text{Percentage Increase} = (1.075 - 1) \times 100\% = 0.075 \times 100\% = 7.5\%
\]
Thus, the percentage increase in the elephant population each year is **7.5%**.
In summary:
- The elephant population in 1990 was **315**.
- The population increases by **7.5%** each year.