To model the decreasing fox population, we need to use the exponential decay formula. The general formula for exponential decay is:
\[ f(x) = P_0 \cdot (1 - r)^x \]
where:
- \( f(x) \) is the population at time \( x \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decrease (as a decimal), and
- \( x \) is the number of years from now.
In this case:
- The initial population \( P_0 = 640 \),
- The rate of decrease \( r = 0.03 \) (which corresponds to 3%).
Thus, we can plug these values into the formula:
\[ f(x) = 640 \cdot (1 - 0.03)^x = 640 \cdot (0.97)^x \]
Now we will estimate the fox population 10 years from now:
\[ f(10) = 640 \cdot (0.97)^{10} \]
Calculating \( (0.97)^{10} \):
\[ (0.97)^{10} \approx 0.737 \]
Now, substituting that back into the equation:
\[ f(10) \approx 640 \cdot 0.737 \approx 471.68 \]
Rounding this to the nearest whole number gives:
\[ f(10) \approx 472 \]
So the final function and estimation is:
The correct option is B: \[ f(x) = 640(0.97)^x \] and the estimated fox population 10 years from now will be approximately 472.