June 2022 Paper 3 Q6

AQACurrent spec9 marksDifferentiationParametric Equations

6 A design for a surfboard is shown in Figure 1.

Figure 1: outline of a surfboard, flat at the left-hand end and rounded at the right-hand end; its length is marked along the bottom and its width (the widest point, towards the right) is marked on the right
Figure 1

The curve of the top half of the surfboard can be modelled by the parametric equations

\[x = -2t^2\]\[y = 9t - 0.7t^2\]

for \(0 \leqslant t \leqslant 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres.

Figure 2: the top half of the surfboard drawn on axes, the curve rising from the left, reaching a maximum, then curving down to the origin O at the right-hand end
Figure 2
(a) Find the length of the surfboard. [2 marks]
(b)
(i) Find an expression for \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [3 marks]
(ii) Hence, show that the width of the surfboard is approximately one third of its length. [4 marks]
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