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IB Maths AI HL 3.2 The modelling process and model critique Question Bank

Practise IB Maths AI HL 2.6 by evaluating model assumptions, domains, predictions and limitations in applied contexts.

Syllabus
First assessment 2021
Course
Mathematics: applications and interpretation HL
Level
HL

Exam points

  • Evaluate whether a modelling assumption is reasonable for the physical or social situation described.
  • Choose a suitable domain and explain why extrapolation or large-input behaviour may be unreliable.
  • Critique a model using residual evidence, context and the consequences of its prediction.

SL 2.6—Modelling process question 1

[Maximum number: 2]

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.

Figure for Question SL 2.6—Modelling process question 1 — IB Maths AI HL

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.

Table for Question SL 2.6—Modelling process question 1 — IB Maths AI HL

Eva first tries to model the height using a linear function, h(t)=a t+b, where a,b∈Ra, b \in \mathbb{R}.

Question (a)

(a)

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, h(t)=pt2+qt+rh(t)=p t^{2}+q t+r, where p,q,r∈Rp, q, r \in \mathbb{R}.

[ 1 ]

Question (b)

(b)

Hence, write down a suitable domain for Eva's function h(t)=pt2+qt+rh(t)=p t^{2}+q t+r.

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.

dV dt=−πR270560h\frac{\mathrm{d} V}{\mathrm{~d} t}=-\pi R^{2} \sqrt{70560 h}
[ 1 ]
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