To model the population of the antelopes, we can use the exponential decay function:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the decay rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P_0 = 570 \)
- \( r = 0.06 \) (since 6% is the decay rate)
The function will be:
\[ P(t) = 570 \cdot (1 - 0.06)^t \]
This simplifies to:
\[ P(t) = 570 \cdot (0.94)^t \]
Now, to find the population after 10 years (\( t = 10 \)):
\[ P(10) = 570 \cdot (0.94)^{10} \]
Calculating \( (0.94)^{10} \):
\[ (0.94)^{10} \approx 0.537 \]
Now substitute this value back into the equation:
\[ P(10) = 570 \cdot 0.537 \approx 306.09 \]
Rounding to the nearest whole number, the number of antelopes after 10 years is:
\[ \boxed{306} \]
So, after 10 years there will be 306 antelopes in the herd.