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.

2 answers

To model the population of the antelopes, we can use the exponential decay function:

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

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.

Given:

  • \( P_0 = 570 \)
  • \( r = 0.06 \) (since 6% is the decay rate)

The function will be:

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

This simplifies to:

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

Now, to find the population after 10 years (\( t = 10 \)):

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

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

\[ (0.94)^{10} \approx 0.537 \]

Now substitute this value back into the equation:

\[ P(10) = 570 \cdot 0.537 \approx 306.09 \]

Rounding to the nearest whole number, the number of antelopes after 10 years is:

\[ \boxed{306} \]

So, after 10 years there will be 306 antelopes in the herd.

Use the image to answer the question.

A coordinate plane's x-axis ranges from negative 2 to 12 and its y-axis ranges from negative 2 to 6, both by 2-unit increments. A solid curve and a dotted line with arrows at both ends are plotted. The solid curve intersects a marked point on the y-axis.

Use the graph of the exponential decay function to determine the equation for the horizontal asymptote of the sequence that is modeled by the graph.

(1 point)
The horizontal asymptote is at y=
.
Remaining Atte