20N.1.SL.TZ0.T_12

pestleMathematicsAISLPaper 120N· sl-2-5-modelling-functionssource ↗

Jean-Pierre jumps out of an airplane that is flying at constant altitude. Before opening his parachute, he goes through a period of freefall.

Jean-Pierre’s vertical speed during the time of freefall, $S$ , in ${m s}^{- 1}$ , is modelled by the following function.

$S (t) = K - 60 (1 . 2^{- t}), t \geq 0$

where $t$ , is the number of seconds after he jumps out of the airplane, and $K$ is a constant. A sketch of Jean-Pierre’s vertical speed against time is shown below.

Jean-Pierre’s initial vertical speed is $0 {m s}^{- 1}$ .

Find the value of $K$ .

[2]

In the context of the model, state what the horizontal asymptote represents.

[1]

Find Jean-Pierre’s vertical speed after $10$ seconds. Give your answer in $km h^{- 1}$ .

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$0 = K - 60 (1 . 2^{0})$ (M1)

Note: Award (M1) for correctly substituted function equated to zero.

$(K =) 60$ (A1) (C2)

[2 marks]

the (vertical) speed that Jean-Pierre is approaching (as $t$ increases) (A1) (C1)
OR
the limit of the (vertical) speed of Jean-Pierre (A1) (C1)

Note: Accept “maximum speed” or “terminal speed”.

[1 mark]

$(S =) 60 - 60 (1 . 2^{- 10})$ (M1)

Note: Award (M1) for correctly substituted function.

$(S =) 50.3096 \dots ({m s}^{- 1})$ (A1)(ft)

Note: Follow through from part (a).

$181 ({km h}^{- 1}) (181.114 \dots ({km h}^{- 1}))$ (A1)(ft) (C3)

Note: Award the final (A1)(ft) for correct conversion of their speed to $km h^{- 1}$ .

[3 marks]

Examiners’ report

[N/A]