To model the population of antelopes with exponential decay, we can use the general formula:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the rate of decrease (expressed as a decimal),
- \( t \) is the time in years.
In this case:
- \( P_0 = 570 \) (the current population),
- \( r = 0.06 \) (the rate of decline of 6%).
Substituting these values into the formula gives:
\[ P(t) = 570 \cdot (1 - 0.06)^t \] \[ P(t) = 570 \cdot (0.94)^t \]
Now, to find the population after 10 years, we will substitute \( t = 10 \):
\[ P(10) = 570 \cdot (0.94)^{10} \]
Now we calculate \( (0.94)^{10} \):
\[ (0.94)^{10} \approx 0.5299 \quad \text{(using a calculator)} \]
Now substitute back into the equation:
\[ P(10) \approx 570 \cdot 0.5299 \approx 302.43 \]
Rounding to the nearest whole number, the population after 10 years will be:
\[ \boxed{302} \]
Thus, there will be approximately 302 antelopes in the herd after 10 years.