IB Maths AI HL Advanced Modelling Questions

The depth of water in a harbour is modelled by the function $D(t)=4\sin \left(\frac{\pi }{6}t\right)+6$, where $D$ is the depth in metres and $t$ is the time in hours after midnight. Find the amplitude, the equation of the principal axis, and the maximum depth of the water.

The population of fish in a newly stocked lake is modelled by the logistic function $P(t)=\frac{1200}{1+19e^{-0.25t}}$, where $t$ is the time in months. Calculate the initial population of fish, and use your Graphic Display Calculator to find the time it takes for the population to reach $600$.

Using a tidal model $D(t)=4\cos \left(\frac{\pi }{6}t\right)+6$, find the exact total amount of time during the first 12 hours ($0\le t\le 12$) when the water depth is strictly greater than $8$ metres.

A radioactive substance follows the decay model $M(t)=200e^{-0.1t}$. Find the initial mass and the mass remaining after exactly 10 years.

The population of a species of insects in a closed environment is modelled by the logistic equation: $$P(t)=\frac{L}{1+C{e}^{-kt}}$$ Initially, the population is 20. After 5 days, the population is 80. The environment can sustain a maximum of 500 insects.
Set up the necessary equations and algebraically solve for the exact values of $L$, $C$, and $k$.