To model the decline of the antelope population, we can use the exponential decay formula:
\[ P(t) = P_0 \cdot e^{kt} \]
Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant (negative for decay),
- \( t \) is time in years.
Given the initial population \( P_0 = 570 \) and a decrease rate of 6% per year, we can find \( k \):
\[ k = -0.06 \]
Now, substituting the values into the formula:
\[ P(t) = 570 \cdot e^{-0.06t} \]
To find the population after 10 years (\( t = 10 \)), we substitute \( t = 10 \) into the equation:
\[ P(10) = 570 \cdot e^{-0.06 \cdot 10} \]
Calculating the exponent:
\[ -0.06 \cdot 10 = -0.6 \]
Now we can calculate \( e^{-0.6} \):
\[ e^{-0.6} \approx 0.5488 \]
Thus, we have:
\[ P(10) = 570 \cdot 0.5488 \approx 312.016 \]
Rounding to the nearest whole number, the population of antelopes after 10 years will be:
\[ \text{After 10 years there will be } \text{312 antelopes.} \]