16N.1.SL.TZ0.T_15

pestleMathematicsAISLPaper 116N· sl-2-5-modelling-functionssource ↗

Gabriella purchases a new car.

The car’s value in dollars, $V$ , is modelled by the function

$V(t) = 12870 - k{(1.1)^t},{\text{ }}t \geqslant 0$

where $t$ is the number of years since the car was purchased and $k$ is a constant.

After two years, the car’s value is $9143.20.

This model is defined for $0 \leqslant t \leqslant n$ . At $n$ years the car’s value will be zero dollars.

Write down, and simplify, an expression for the car’s value when Gabriella purchased it.

[2]

Find the value of $k$ .

[2]

Find the value of $n$ .

[2]

Markscheme / solution

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

$12870 - k{(1.1)^0}$ (M1)

Note: Award (M1) for correct substitution into $V(t)$ .

$= 12870 - k$ (A1) (C2)

Note: Accept $12870 - 3080$ OR 9790 for a final answer.

[2 marks]

$9143.20 = 12870 - k{(1.1)^2}$ (M1)

Note: Award (M1) for correct substitution into $V(t)$ .

$(k = ){\text{ }}3080$ (A1) (C2)

[2 marks]

$12870 - 3080{(1.1)^n} = 0$ (M1)

Note: Award (M1) for correct substitution into $V(t)$ .

N16/5/MATSD/SP1/ENG/TZ0/15.c/M (M1)

Note: Award (M1) for a correctly shaped curve with some indication of scale on the vertical axis.

$(n = ){\text{ }}15.0{\text{ }}(15.0033 \ldots )$ (A1)(ft) (C2)

Note: Follow through from part (b).

[2 marks]

Examiners’ report

[N/A]