A-level · Coordinate geometry

Modelling with parametric equationsExam-style questions

Using parametric equations to model real situations where x and y each depend on time — projectiles and rotating points — reading positions, maxima and ranges from the model, and finding the Cartesian equation of the path.

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19 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Figure 1 shows the design for the cross-section of a wing for a model aircraft.

Figure 1The front edge of the cross-section is at $A$ and the back edge is at $B$ . The cross-section is symmetrical about the straight line $AB$ .

The upper edge of the cross-section is modelled by the curve with parametric equations

$x = 20(1 - \cos t) \qquad y = 2\sin t + \sin 2t$

for $0 \leq t \leq \pi$ , where $x$ and $y$ are measured in centimetres. The points $A$ and $B$ are where $t = 0$ and $t = \pi$ respectively, as shown in Figure 2.

Figure 2(a) Find the length $AB$

(2 marks)
(b) (i) Find an expression for $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$

(ii) Hence, show that the greatest thickness of the cross-section is approximately one eighth of the length $AB$

(7 marks)
Level 4 · 9 marksTry it
2
A furniture maker is designing a rocking chair, shown in Figure 1.

Figure 1Each runner of the chair is shaped like the curve with parametric equations

$x = 20t \qquad y = 2t^2 + \frac{32}{t} \qquad 1 \leq t \leq 4$

where $x$ and $y$ are lengths in centimetres. Figure 2 shows one runner drawn on a pair of axes.

The ends of the runner, $P$ and $Q$ , are the points where $t = 1$ and $t = 4$ respectively.

Figure 2(a) Show that a Cartesian equation of the curve is

$y = \frac{x^2}{200} + \frac{640}{x}$

(2 marks)
(b) (i) Find an expression for $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$

(ii) The chair rests on horizontal ground, with each runner touching the ground at its lowest point.

Hence, find the height of $Q$ above the ground.

(6 marks)
(c) Suggest one limitation of the model.

(1 mark)
Level 4 · 9 marksTry it