Question
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.(1 point) After 10 years there will be antelopes.
Answers
GPT-4o mini
To create an exponential decay function to model the population of the antelopes, we can use the formula:
\[
P(t) = P_0 e^{kt}
\]
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.
Given that the population declines at a rate of 6% per year, we can express the decay constant \(k\) as:
\[
k = -0.06
\]
The initial population \(P_0\) is given as 570.
Therefore, the function modeling the population becomes:
\[
P(t) = 570 e^{-0.06t}
\]
Next, we want to find the population after 10 years, so we substitute \(t = 10\) into the function:
\[
P(10) = 570 e^{-0.06 \times 10}
\]
\[
P(10) = 570 e^{-0.6}
\]
Now we can calculate \(e^{-0.6}\):
Using a calculator, \(e^{-0.6} \approx 0.5488\).
Now we substitute this back into the equation:
\[
P(10) \approx 570 \times 0.5488 \approx 312.996
\]
Rounding to the nearest whole number, we find:
\[
P(10) \approx 313
\]
Thus, after 10 years there will be approximately **313 antelopes** in the herd.
\[
P(t) = P_0 e^{kt}
\]
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.
Given that the population declines at a rate of 6% per year, we can express the decay constant \(k\) as:
\[
k = -0.06
\]
The initial population \(P_0\) is given as 570.
Therefore, the function modeling the population becomes:
\[
P(t) = 570 e^{-0.06t}
\]
Next, we want to find the population after 10 years, so we substitute \(t = 10\) into the function:
\[
P(10) = 570 e^{-0.06 \times 10}
\]
\[
P(10) = 570 e^{-0.6}
\]
Now we can calculate \(e^{-0.6}\):
Using a calculator, \(e^{-0.6} \approx 0.5488\).
Now we substitute this back into the equation:
\[
P(10) \approx 570 \times 0.5488 \approx 312.996
\]
Rounding to the nearest whole number, we find:
\[
P(10) \approx 313
\]
Thus, after 10 years there will be approximately **313 antelopes** in the herd.