The population, \(P\), of a species of bird on a remote island is modeled by the logistic function:
\(P(t) = \frac{L}{1+C\,e^{-kt}}\),
where \(t\) is the number of years since the observation began, \(L=2000\) is the carrying capacity, and \(C\) and \(k\) are positive constants. It is given that initially (\(t=0\)) \(P(0)=250\) and after 5 years \(P(5)=600\).
(a) Calculate the value of \(C\). [2]
(b) Show that \(k \approx 0.218\). [3]
(c) Predict the bird population after 15 years. [2]
(d) Find the number of years it will take for the population to reach 1800 birds. [1]
Question
Calculate the value of \(C\). [2]
Hint
Hint
Hint
Substitute \(t=0\) in \(P(t)=\frac{2000}{1+C\,e^{-kt}}\) to get \(250=\frac{2000}{1+C}\) and solve for \(C\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Show that \(k \approx 0.218\). [3]
Hint
Hint
Hint
Substitute \(t = 5\) with \(P(5)=600\) into the model and solve for \(k\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
k = ln(3)/5 ≈ 0.218
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Predict the bird population after 15 years. [2]
Hint
Hint
Hint
Substitute \(t = 15\) and the value of \(k\) into the logistic model and evaluate.
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
Approximately 1600 birds
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
Question
Find the number of years it will take for the population to reach 1800 birds. [1]
Hint
Hint
Hint
Set \(P(t)=1800\) in the model and solve for \(t\).
-
Answer
Answer
-
Mark Scheme
Mark Scheme
-
Explanation
Explanation
Complete Answer
Approximately 21.4 years
-
Mark Scheme
Mark Scheme
-
Answer Explanation
Answer Explanation
