Question
The current fox population in a protected region is 367 and decreasing by 8% each year. What exponential function models the fox population?
A. F(x) = 367 * 1.08^x
B. F(x) = 367 * 0.92^x
C. F(x) = 0.98 * 367^z
D. F(x) = 1.08 * 367^z
A. F(x) = 367 * 1.08^x
B. F(x) = 367 * 0.92^x
C. F(x) = 0.98 * 367^z
D. F(x) = 1.08 * 367^z
Answers
GPT-4o mini
To model the fox population that is currently 367 and decreasing by 8% each year, we can use the formula for exponential decay:
\[ F(x) = F_0 \times (1 - r)^x \]
where:
- \( F_0 \) is the initial amount (the current population),
- \( r \) is the rate of decrease (as a decimal),
- \( x \) is the number of years.
In this case:
- The initial population \( F_0 = 367 \),
- The rate of decrease \( r = 0.08 \) (which corresponds to an 8% decrease).
Thus, \( (1 - r) = 1 - 0.08 = 0.92 \).
Now substituting these values into the formula gives:
\[ F(x) = 367 \times 0.92^x \]
Therefore, the correct exponential function that models the fox population is:
**B. \( F(x) = 367 \times 0.92^x \)**
\[ F(x) = F_0 \times (1 - r)^x \]
where:
- \( F_0 \) is the initial amount (the current population),
- \( r \) is the rate of decrease (as a decimal),
- \( x \) is the number of years.
In this case:
- The initial population \( F_0 = 367 \),
- The rate of decrease \( r = 0.08 \) (which corresponds to an 8% decrease).
Thus, \( (1 - r) = 1 - 0.08 = 0.92 \).
Now substituting these values into the formula gives:
\[ F(x) = 367 \times 0.92^x \]
Therefore, the correct exponential function that models the fox population is:
**B. \( F(x) = 367 \times 0.92^x \)**