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 (negative for decay),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given that the herd is declining at a rate of 6% per year, we can express the decay constant \( k \) as:
\[ k = -0.06 \]
The initial population \( P_0 \) is 570. Therefore, the function can be written as:
\[ P(t) = 570 \cdot e^{-0.06t} \]
To find the population after 10 years, we will substitute \( t = 10 \) into the function:
\[ P(10) = 570 \cdot e^{-0.06 \cdot 10} \] \[ P(10) = 570 \cdot e^{-0.6} \]
Calculating \( e^{-0.6} \):
\[ e^{-0.6} \approx 0.5488 \]
Now substitute this value back into the equation:
\[ P(10) \approx 570 \cdot 0.5488 \approx 312.42 \]
Rounding to the nearest whole number, we find:
\[ P(10) \approx 312 \]
Therefore, after 10 years there will be approximately 312 antelopes in the herd.