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.
After 10 years there will be ____ antelopes.

Answer 1

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 (570 antelopes),
- \( k \) is the decay constant,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.

Since the population is declining at a rate of 6% per year, we have:

\[
k = -0.06
\]

Now, substituting \( P_0 = 570 \) and \( k = -0.06 \) into the formula gives us:

\[
P(t) = 570 \cdot e^{-0.06t}
\]

Now we want to find the population after 10 years, so we set \( t = 10 \):

\[
P(10) = 570 \cdot e^{-0.06 \times 10}
\]

Calculating \( -0.06 \times 10 \):

\[
-0.06 \times 10 = -0.6
\]

Thus, we have:

\[
P(10) = 570 \cdot e^{-0.6}
\]

Now we compute \( e^{-0.6} \):

\[
e^{-0.6} \approx 0.5488
\]

Now substituting back into the equation:

\[
P(10) \approx 570 \cdot 0.5488 \approx 312.696
\]

Rounding to the nearest whole number, we find:

\[
P(10) \approx 313
\]

Therefore, after 10 years, there will be approximately **313** antelopes in the herd.