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)

Question

After 10 years there will be  antelopes.

Answer 1

To model the decline of the antelope herd, 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,
• \( t \) is the time in years,
• \( e \) is the base of the natural logarithm.

However, since the problem specifies a constant percentage decrease, we can use a slightly different formula:

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

where:

• \( r \) is the rate of decay expressed as a decimal.

For this specific problem:

• \( P_0 = 570 \) (the initial population),
• \( r = 0.06 \) (the decay rate of 6%),
• and \( t = 10 \) years.

So, our function becomes:

\[ P(t) = 570 (1 - 0.06)^{t} \] \[ P(t) = 570 (0.94)^{t} \]

Now we can calculate the population after 10 years:

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

Calculating \( (0.94)^{10} \):

\[ (0.94)^{10} \approx 0.527 \]

Now multiply that by 570:

\[ P(10) \approx 570 \times 0.527 \] \[ P(10) \approx 300.39 \]

Rounding to the nearest whole number:

After 10 years, there will be approximately 300 antelopes in the herd.