Question
The elephant population of a nature preserve since 1990 is modeled by the exponential function f(x)=315⋅1.075
x
. Find the elephant population in 1990 and the percentage at which the population increases each year.
A. The elephant population in 1990 is 315, and the population increases by 1.075% each year. B. The elephant population in 1990 is 315, and the population increases by 7.5% each year. C. The elephant population in 1990 is 315, and the population increases by 0.75% each year. D. The elephant population in 1990 is 339, and the population increases by 7.5% each year.
x
. Find the elephant population in 1990 and the percentage at which the population increases each year.
A. The elephant population in 1990 is 315, and the population increases by 1.075% each year. B. The elephant population in 1990 is 315, and the population increases by 7.5% each year. C. The elephant population in 1990 is 315, and the population increases by 0.75% each year. D. The elephant population in 1990 is 339, and the population increases by 7.5% each year.
Answers
GPT-4o mini
To analyze the exponential function given by \( f(x) = 315 \cdot 1.075^x \):
1. **Finding the elephant population in 1990:**
In the context of this function, \( x \) typically represents the number of years since a baseline year, which we can assume is 1990. If we set \( x = 0 \) for the year 1990, we have:
\[
f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315.
\]
Therefore, the elephant population in 1990 is \( 315 \).
2. **Calculating the percentage increase each year:**
The term \( 1.075 \) is the growth factor. To convert this to a percentage increase, we can subtract 1 and then convert it to a percentage:
\[
\text{Percentage increase} = (1.075 - 1) \times 100\% = 0.075 \times 100\% = 7.5\%.
\]
Now that we have established both points:
- The elephant population in 1990 is **315**.
- The population increases by **7.5%** each year.
Thus, the correct answer is **B. The elephant population in 1990 is 315, and the population increases by 7.5% each year.**
1. **Finding the elephant population in 1990:**
In the context of this function, \( x \) typically represents the number of years since a baseline year, which we can assume is 1990. If we set \( x = 0 \) for the year 1990, we have:
\[
f(0) = 315 \cdot 1.075^0 = 315 \cdot 1 = 315.
\]
Therefore, the elephant population in 1990 is \( 315 \).
2. **Calculating the percentage increase each year:**
The term \( 1.075 \) is the growth factor. To convert this to a percentage increase, we can subtract 1 and then convert it to a percentage:
\[
\text{Percentage increase} = (1.075 - 1) \times 100\% = 0.075 \times 100\% = 7.5\%.
\]
Now that we have established both points:
- The elephant population in 1990 is **315**.
- The population increases by **7.5%** each year.
Thus, the correct answer is **B. The elephant population in 1990 is 315, and the population increases by 7.5% each year.**