IB Maths AI HL 2.9 Further modelling functions

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George is researching the growth in the number of electric vehicles (EVs) in the European Union in order to investigate some of the difficulties that might arise if the target of banning sales of all petrol and diesel cars in 2035 is to be met.
George begins his research by predicting how many electric vehicles (EVs) will be in the European Union in 2035.
The number of EVs in the European Union, N, measured in millions, is shown in the following table. The time t is measured in years from the beginning of 2016 where $t \in \mathbb{R}$ .

Table for Question AHL 2.9 (HL)—Further modelling functions question 1 — IB Maths AI HL

George models this data set using the logistic function $N=\frac{310}{1+C \mathrm{e}^{-k t}}$ , where $C, k \in \mathbb{R}^{+}$ .

(a)

In context, explain the significance of 310 in George's logistic function.

[ 1 ]

(b)

Use the value of N at t=1 to show that $C=1549 \mathrm{e}^{k}$ .

[ 2 ]

correct substitution $0.2=\frac{310}{1+C \mathrm{e}^{-k}}$
correct rearrangement which leads to correct final answer
eg $1+C \mathrm{e}^{-k}=1550$ OR $C \mathrm{e}^{-k}=1549$ OR $0.2 C \mathrm{e}^{-k}=309.8$ $C=1549 \mathrm{e}^{k}$

(c)

Use the value of N at t=7 to find a second expression for C.

[ 3 ]

correct substitution $3.1=\frac{310}{1+C \mathrm{e}^{-7 k}}$
correct first stage of rearrangement
eg $1+C \mathrm{e}^{-7 k}=100$ OR $3.1+3.1 \mathrm{Ce}^{-7 k}=310$ $C=99 \mathrm{e}^{7 k}$

Note: Allow any correct expression eg $C=\frac{99}{\mathrm{e}^{-7 k}}$ or $C=\frac{306.9}{3.1 \mathrm{e}^{-7 k}}$ .

(d)

Use your answers to part (b) to find a value for

[ 3 ]

(i)

$\quad k$ .

[ 2 ]

(c) valid method for finding either variable
eg graphically or elimination of $C, \mathrm{e}^{6 k}=\frac{1549}{99}$
(i) $k=0.458(0.458374 \ldots)$

Note: Award at most M1A0 if values other than at t=1 or t=7 are used.

(ii)

C.

George uses his model to predict values of N at the end of the years 2017 to 2021. These values are shown in the following table, correct to one decimal place.

Table for Question (ii) — IB Maths AI HL

[ 1 ]

$C=2450(2449.74 \ldots)$

Note: Do not award A1FT on a wrong value for k if the method used is not clearly shown.

Note: Accept $C=2440(2443.33 \ldots)$ from substitution of 3 sf value for k in $C=99 \mathrm{e}^{7 k}$

Note: Award AO if values other than at t=1 or t=7 are used.

\left(N=\frac{310}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots t}}\right)

(e)

Calculate the value of a correct to one decimal place.

[ 2 ]

substitution of t=4 into their logistic function

\begin{aligned} & N=\frac{310}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots \times 4}} \\ & a=0.789606 \ldots \end{aligned}

Note: If using 3 sf values for k and $C: a=0.788346 \ldots$ . If using C=2440 then a=0.792.

a=0.8

Note: The answer must be given correct to 1 decimal place

(f)

Use the model to predict the number of EVs in the European Union at the end of 2035.

One of the main difficulties in meeting the European Union's target is the provision of sufficient public charging points.

The European Union estimates that 80 % of car owners will be able to charge their cars at home and the remaining 20 % will require public charging points. It is planned to have one public charging point for every 10 cars that require them.

[ 2 ]

(g)

Use George's model to find an expression for the total number of public charging points required at time t.

At the end of 2020 there were 0.22 million public charging points in the European Union and at the end of 2022 there were 0.54 million.

[ 2 ]

multiplication of their model by one of 0.2 or 0.1

Note: For the M1 ignore any extra incorrect multipliers.
number of public charging points needed in year t is

\begin{gathered} \frac{310}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots t}} \times 0.2 \times 0.1\left(=\frac{6.2}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots t}}\right) \text { (million) OR } \\ \frac{310,000,000}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots t}} \times 0.2 \times 0.1\left(=\frac{6,200,000}{1+2449.74 \ldots \mathrm{e}^{-0.458374 \ldots t}}\right) \end{gathered}

IB Maths AI HL 2.9 Further modelling functions Question Bank

Exam points

AHL 2.9 (HL)—Further modelling functions question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (i)

Question (ii)

Question (e)

Question (f)

Question (g)