IB Mathematics Applications & Interpretation Question Type 5: Determining Parameter Values Based on Information on the Logistic Model Exercises

Fit a logistic model $P(t)=\frac{L}{1+C{e}^{-kt}}$ to data: $P(0)=5$ and $P(3)=20$, given carrying capacity $L=100$. Determine $C$ and $k$.

Find the constant $C$ in the logistic function $P(t)=\frac{100}{1+C{e}^{-0.3t}},$ if the initial value is $P(0)=120$.

Given a logistic growth model $$P(t)=\frac{50}{1+C{e}^{-0.6t}},$$ and the initial population is $P(0)=10$, determine the value of $C$.

Write the logistic function for a population of bacteria with carrying capacity $L=8$, initial size $P(0)=2$, and growth constant $k=0.5$.

Formulate the logistic model $P(t)=\frac{L}{1+C{e}^{-rt}}$ given $L=10$, initial $P(0)=12$, and intrinsic growth rate $r=3$.

A logistic model has $L=200$, $P(0)=10$ and $P(5)=50$. Determine the growth constant $k$ in

$$P(t)=\frac{200}{1+C{e}^{-kt}}$$

A species has carrying capacity $L=9$ and initial population $P(0)=10$ following a logistic model $$P(t)=\frac{9}{1+C{e}^{-kt}}.$$ Compute $C$.

For a logistic model with $L=500$, $P(0)=50$, and $r=0.4$, find $P(4)$.

A logistic curve with carrying capacity $1000$ has growth constant $0.1$ and initial population $100$. How long until the population reaches half the carrying capacity?

Given $P(t)=\frac{1000}{1+9e^{-0.2t}}$, find the time $t$ at which $P(t)=800$.

Given the logistic model $$P(t)=\frac{20}{1+4e^{-0.3t}},$$ compute $P(5)$.

A logistic population has a carrying capacity $L=200$ and initial population $P(0)=20$. If the population reaches $P(2)=80$, find the growth rate $k$.