To model the decline of the antelope population using an exponential decay function, we can use the 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, and
- \( t \) is the time in years.
In this case, we start with an initial population of \( P_0 = 570 \) antelopes and a decay rate of \( 6% \) per year, which means \( k = -0.06 \) (the negative sign reflects decay).
Thus, the function becomes:
\[ P(t) = 570 \cdot e^{-0.06t} \]
To find the population after 10 years, we substitute \( t = 10 \) into the equation:
\[ P(10) = 570 \cdot e^{-0.06 \cdot 10} \] \[ P(10) = 570 \cdot e^{-0.6} \]
Next, we calculate \( e^{-0.6} \):
\[ e^{-0.6} \approx 0.5488 \] (using a calculator)
Now, plug this value back into our equation:
\[ P(10) \approx 570 \cdot 0.5488 \] \[ P(10) \approx 312.396 \]
Rounding this to the nearest whole number, we find:
After 10 years, there will be approximately 312 antelopes in the herd.