Question

In a certain geographic location, a herd of antelopes is declining at a rate of six percent every year. If there are
currently 570 antelopes in the herd, create an exponential decay function to model the problem. If the decrease
in population continues at this rate, how many antelopes will be in the herd after 10 years? Round your answer
to the nearest whole number

Answer 1

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:

\[
\boxed{313}
\]