IB Maths AI SL AI 2.2 Modelling questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper
Paper 1Paper 2
Difficulty
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Question 1

Paper 1Medium14 marks

A horticulture centre monitors the depth of water in a shallow pond during a controlled fill-and-drain test. Let tt be the time, in hours, since the test began, and let hh be the depth of water, in centimetres.

tt0257
hh1.37.610.17.0
(a)

Choose the most suitable model from a linear, quadratic or exponential model for these data. Justify your choice.

[2]
(b)

Using only the observations at t=0t=0 t=2t=2 and t=5t=5 fit a quadratic model of the form h=at2+bt+ch=at^2+bt+c Show that

h=−0.463t2+4.08t+1.30,h=-0.463t^2+4.08t+1.30,

where the coefficients are given to 3 significant figures. Retain unrounded values for later parts.

[3]
(c)

Use your fitted model to estimate the depth at t=7t=7 Compare your estimate with the recorded depth.

[2]
(d)

Use your fitted model to estimate the time at which the depth is 88 cm on the falling part of the curve. Give your answer to the nearest 0.010.01 hour.

[3]
(e)

State a reasonable domain for using this model, based on the observations. Use the model to estimate the depth at t=10t=10 and comment on whether this prediction is reasonable.

[4]

Question 2

Paper 1Medium14 marks

An engineer records the water height, hh cm, in a reservoir while a pump fills it. The pump remains on throughout a planned 8-minute run, and there is no outlet. The measurements shown in Figure 1 were taken during the first 5 minutes. Heights are recorded to the nearest 0.10.1 cm.

012345678051015202530t (minutes)h (cm)(0, 3.8)(1, 10.3)(2, 15.6)(3, 20.0)(4, 23.1)(5, 25.0)
(a)

Using the shape of the data in Figure 1, choose a linear or quadratic model for hh in terms of tt Justify your choice.

[2]
(b)

Let h=at2+bt+ch=at^2+bt+c Using the measurements at t=0t=0 t=2t=2 and t=4t=4 show that a fitted model is

h=−0.5375t2+6.975t+3.8.h=-0.5375t^2+6.975t+3.8.
[4]
(c)

Hence, using this model, determine the first time during the 5-minute measurement interval at which the model predicts h=25.0h=25.0 cm. Give your answer to the nearest 0.010.01 minute.

[3]
(d)

At t=3t=3 compare the model’s predicted height with the recorded height. State what this comparison suggests about the model’s fit at that time.

[2]
(e)

Use the fitted model to predict the height after 8 minutes. State whether this prediction should be considered reliable, giving two reasons.

[3]

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