To model the decline of the antelope population, we can use the exponential decay function:
\[ P(t) = P_0 e^{kt} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
First, we translate the percentage decline rate into a decay constant. A decrease of 6% per year means that the remaining population each year is 94% of the previous year's population, which can be modeled as:
\[ P(t) = 570 \times (0.94)^t \]
Here, \( P_0 = 570 \) and the decay factor \( (0.94) \) is derived from \( 1 - 0.06 = 0.94 \).
To find the population after 10 years, we substitute \( t = 10 \) into the function:
\[ P(10) = 570 \times (0.94)^{10} \]
Now we can calculate \( (0.94)^{10} \):
\[ (0.94)^{10} \approx 0.527 \]
Now multiply by the initial population:
\[ P(10) \approx 570 \times 0.527 \approx 300.39 \]
Rounding to the nearest whole number, we find:
\[ \text{After 10 years, there will be } \approx 300 \text{ antelopes.} \]
Thus, the final answer is:
After 10 years there will be 300 antelopes.