Question
The current fox population in a protected region is 367 and decreasing by 8% each year. What exponential function models the fox population?(1 point)
Responses
f(x) = 367 ⋅1.08x
𝑓
(
𝑥
)
=
367
⋅
1.08
𝑥
f(x) = 367 ⋅1.08x
𝑓
(
𝑥
)
=
367
⋅
1.08
𝑥
f(x) = 367 ⋅0.92x
𝑓
(
𝑥
)
=
367
⋅
0.92
𝑥
f(x) = 367 ⋅0.92x
𝑓
(
𝑥
)
=
367
⋅
0.92
𝑥
f(x) = 0.98 ⋅367z
𝑓
(
𝑥
)
=
0.98
⋅
367
𝑧
f(x) = 0.98 ⋅367z
𝑓
(
𝑥
)
=
0.98
⋅
367
𝑧
Responses
f(x) = 367 ⋅1.08x
𝑓
(
𝑥
)
=
367
⋅
1.08
𝑥
f(x) = 367 ⋅1.08x
𝑓
(
𝑥
)
=
367
⋅
1.08
𝑥
f(x) = 367 ⋅0.92x
𝑓
(
𝑥
)
=
367
⋅
0.92
𝑥
f(x) = 367 ⋅0.92x
𝑓
(
𝑥
)
=
367
⋅
0.92
𝑥
f(x) = 0.98 ⋅367z
𝑓
(
𝑥
)
=
0.98
⋅
367
𝑧
f(x) = 0.98 ⋅367z
𝑓
(
𝑥
)
=
0.98
⋅
367
𝑧
Answers
GPT-4o mini
To model the fox population that is decreasing by 8% each year, we need to use an exponential decay function. The general form of such a function is:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years.
In this case:
- \( P_0 = 367 \)
- The rate of decline \( r = 0.08 \), so \( (1 - r) = 0.92 \).
Thus, the function modeling the population would be:
\[ f(x) = 367 \cdot 0.92^x \]
Therefore, the correct choice from the options provided is:
**f(x) = 367 ⋅ 0.92^x**
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the rate of decline (as a decimal),
- \( x \) is the number of years.
In this case:
- \( P_0 = 367 \)
- The rate of decline \( r = 0.08 \), so \( (1 - r) = 0.92 \).
Thus, the function modeling the population would be:
\[ f(x) = 367 \cdot 0.92^x \]
Therefore, the correct choice from the options provided is:
**f(x) = 367 ⋅ 0.92^x**