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.
Since the population is declining at a rate of six percent per year, we have:
\[ k = 0.06 \]
The initial population \( P_0 \) is given as 570. Hence, the decay function can be expressed as:
\[ P(t) = 570 \cdot e^{-0.06t} \]
However, we can also use a simpler form of the exponential decay function, which is based on the formula for percentage decay:
\[ P(t) = P_0 \cdot (1 - r)^t \]
where \( r \) is the rate of decay (0.06 in this case). Thus, the function becomes:
\[ P(t) = 570 \cdot (1 - 0.06)^t \] \[ P(t) = 570 \cdot (0.94)^t \]
Now, to find the population after 10 years (\( t = 10 \)), we can substitute \( t \) with 10:
\[ P(10) = 570 \cdot (0.94)^{10} \]
Calculating \( (0.94)^{10} \):
\[ (0.94)^{10} \approx 0.5220 \quad (\text{using a calculator}) \]
Now substituting this value back into the equation:
\[ P(10) = 570 \cdot 0.5220 \approx 297.54 \]
Rounding to the nearest whole number, we get:
\[ P(10) \approx 298 \]
Thus, after 10 years, there will be approximately 298 antelopes in the herd.