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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
antelopes.

3 answers

To model the decline of the antelope population, we can use an exponential decay function of the form:

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

where:

  • \( P(t) \) is the population after time \( t \),
  • \( P_0 \) is the initial population,
  • \( k \) is the decay constant (a negative value since it's a decay), and
  • \( t \) is the time in years.

Given that the initial population \( P_0 = 570 \) and the population is declining at a rate of 6% per year, we can express the decay as:

\[ k = -0.06 \]

Now we can substitute \( P_0 \) and \( k \) into the formula:

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

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

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

We can calculate \( e^{-0.6} \) (using a calculator or a reference for \( e^x \)):

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

Now plug this value back into the equation:

\[ P(10) \approx 570 \cdot 0.5488 \] \[ P(10) \approx 312.81 \]

Rounding to the nearest whole number, we find:

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

wrong

I apologize for any mistakes made in my previous response. Let's redo the calculation step by step.

  1. Set up the exponential decay function: The general form for exponential decay is:

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

    where:

    • \( P(t) \) is the population after \( t \) years,
    • \( P_0 \) is the initial population,
    • \( r \) is the decay rate (as a decimal),
    • \( t \) is the time in years.

    Here, we have:

    • \( P_0 = 570 \) (the current population),
    • \( r = 0.06 \) (6% decline).

    Therefore, the function will be:

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

  2. Calculate the population after 10 years: Now we need to calculate \( P(10) \):

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

    First, calculate \( (0.94)^{10} \):

    \( (0.94)^{10} \approx 0.5220 \) (using a calculator or exponentiation).

    Now substitute that into the equation:

    \[ P(10) \approx 570 \cdot 0.5220 \approx 297.54 \]

  3. Round to the nearest whole number: Rounding \( 297.54 \) gives us approximately 298.

So, after 10 years, there will be approximately 298 antelopes in the herd. Thank you for your patience, and I appreciate your understanding!