19N.1.SL.TZ0.T_11

pestleMathematicsAISLPaper 119N· sl-2-5-modelling-functionssource ↗

Elvis Presley is an extremely popular singer. Although he passed away in $1977$ , many of his fans continue to pay tribute by dressing like Elvis and singing his songs.

The number of Elvis impersonators, $N (t)$ , can be modelled by the function

$N (t) = 170 \times 1 . 31^{t},$

where $t$ , is the number of years since $1977$ .

Write down the number of Elvis impersonators in $1977$ .

[1]

Calculate the time taken for the number of Elvis impersonators to reach $130 000$ .

[2]

Calculate the number of Elvis impersonators when $t = 70$ .

[2]

The world population in $2047$ is projected to be $9 500 000 000$ people.

Use this information to explain why the model for the number of Elvis impersonators is unrealistic.

[1]

Markscheme / solution

$170$ (A1) (C1)

[1 mark]

$130 000 = 170 \times 1 . 31^{t}$ (M1)

Note: Award (M1) for equating $130 000$ to the exponential function.

$(t =) 24.6$ ( $24.5882 \dots$ (years)) (A1) (C2)

[2 marks]

$170 \times 1 . 31^{70}$ (M1)

Note: Award (M1) for correct substitution in the function $N (t)$ .

$2.75 \times 10^{10} (2.75067 \dots \times 10^{10}, 27 500 000 000, 27 506 771 343)$ (A1) (C2)

[2 marks]

The number of Elvis impersonators in $2047$ , is greater than the world population. (R1) (C1)

$2.75 \times 10^{10} > 9 500 000 000$ (R1) (C1)

Note: Award (R1) for a correct comparison of their number of impersonators with the world population. Follow through from part (c) if a reasonable argument can be made that the model is unrealistic.
Award (R0) if the number of impersonators is not explicitly seen in part (c) or in part (d).

[1 mark]

Examiners’ report

[N/A]