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 population, we can use the exponential decay function of the form:

\[ P(t) = P_0 e^{kt} \]

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.

In this case:

• The initial population \( P_0 = 570 \).
• The decay rate is 6% per year, which means \( k = -0.06 \).

Thus, the population function can be written as:

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

Now, to find the population after 10 years, we can substitute \( t = 10 \) into the function:

\[ P(10) = 570 e^{-0.06 \cdot 10} \] \[ P(10) = 570 e^{-0.6} \]

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

Using a calculator, we find:

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

Now, plug this value back into the equation:

\[ P(10) = 570 \times 0.5488 \] \[ P(10) \approx 312.996 \]

Rounding this to the nearest whole number, we find:

After 10 years, there will be approximately 313 antelopes in the herd.