IB Maths AI HL Sl 2 6 Modelling Process Topic Practice

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Study SL 2.6—Modelling process conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

Question 1

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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 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 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}.

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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}
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