To create an exponential decay function for the antelope population, we can use the general form of the exponential decay function:
\[ P(t) = P_0 \cdot e^{kt} \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( k \) is the decay constant,
- \( t \) is time in years,
- \( e \) is the base of the natural logarithm.
In this case, the initial population \( P_0 = 570 \) and the decay rate is 6% per year. Since the population is decreasing, \( k \) will be negative:
\[ k = -0.06 \]
Thus, the exponential decay function for the antelope population is:
\[ P(t) = 570 \cdot e^{-0.06t} \]
To find the population after 10 years, we need to evaluate \( P(10) \):
\[ P(10) = 570 \cdot e^{-0.06 \cdot 10} \]
Calculating the exponent:
\[ -0.06 \cdot 10 = -0.6 \]
Now, calculate \( e^{-0.6} \):
\[ e^{-0.6} \approx 0.5488 \]
Now substitute this value back into the population function:
\[ P(10) = 570 \cdot 0.5488 \approx 312.996 \]
Rounding to the nearest whole number, the population after 10 years is:
\[ \boxed{313} \]