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.(1 point)
After 10 years there will be ___ antelope

Answer 1

To model the decline of the antelope population, we can use the exponential decay 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 time in years.

Given the initial population \( P_0 = 570 \) and a decrease rate of 6% per year, we can find \( k \):

\[
k = -0.06
\]

Now, substituting the values into the formula:

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

To find the population after 10 years (\( t = 10 \)), we substitute \( t = 10 \) into the equation:

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

Calculating the exponent:

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

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

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

Thus, we have:

\[
P(10) = 570 \cdot 0.5488 \approx 312.016
\]

Rounding to the nearest whole number, the population of antelopes after 10 years will be:

\[
\text{After 10 years there will be } \text{312 antelopes.}
\]