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 model the decline of the antelope population using an exponential decay function, we can use the formula:

\[
P(t) = P_0 \cdot (1 - r)^t
\]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the decay rate (as a decimal),
- \( t \) is the time in years.

In this case:
- \( P_0 = 570 \) (the initial population of antelopes),
- \( r = 0.06 \) (the decay rate of 6% per year),
- \( t = 10 \) (the number of years).

Substituting the values into the exponential decay function, we get:

\[
P(t) = 570 \cdot (1 - 0.06)^{10}
\]

Calculating \( 1 - 0.06 \):

\[
1 - 0.06 = 0.94
\]

Now substituting this into the equation:

\[
P(10) = 570 \cdot (0.94)^{10}
\]

Now we need to calculate \( (0.94)^{10} \):

\[
(0.94)^{10} \approx 0.5220 \quad (\text{Using a calculator})
\]

Now, we can calculate the estimated population after 10 years:

\[
P(10) = 570 \cdot 0.5220 \approx 298.74
\]

Rounding to the nearest whole number:

\[
P(10) \approx 299
\]

So, the expected number of antelopes in the herd after 10 years is approximately **299**.