Question 1

Paper 1, no calculatorHard6 marks

A Ferris-wheel passenger’s height above the ground, hh metres, tt minutes after the ride begins is modelled by

h(t)=asin⁡(b(t+c))+d,h(t)=a\sin\bigl(b(t+c)\bigr)+d,

where a,b>0a,b>0 and 0≤c<2πb0\leq c<\frac{2\pi}{b}

The passenger’s lowest and highest heights are 33 m and 1515 m respectively. The wheel reaches its highest point at t=2t=2 its lowest point at t=8t=8 and its next highest point at t=14t=14

(a)

Write down the amplitude of hh

[1]
(b)

Use the model and observations above.

(i)

Show that

h(t)=6sin⁡(π6(t+1))+9.h(t)=6\sin\left(\frac{\pi}{6}(t+1)\right)+9.
[3]
(ii)

Hence determine all values of tt where 0≤t≤280\leq t\leq 28 for which the passenger’s height is 1212 m.

[2]

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