Practise defining variables, units, domains and initial conditions, fitting models, testing predictions and explaining assumptions or extrapolation limits.
This question explores models for the height of water in a cylindrical container as water drains out.
The diagram shows a cylindrical water container of height 3.2 metres and base radius 1 metre. At the base of the container is a small circular valve, which enables water to drain out.
Eva closes the valve and fills the container with water.
At time t=0, Eva opens the valve. She records the height, h metres, of water remaining in the container every 5 minutes.
Eva first tries to model the height using a linear function, h(t)=a t+b, where .
Suggest why Eva's use of the linear regression equation in this way could be unreliable.
Eva thinks she can improve her model by using a quadratic function, , where .
Hence, write down a suitable domain for Eva's function .
Let V be the volume, in cubic metres, of water in the container at time t minutes.
Let R be the radius, in metres, of the circular valve.
Eva does some research and discovers a formula for the rate of change of V.