(a)     The circular Ferris wheel has a radius of 10 metres and is revolving at a rate of 3 radians per minute. Determine how fast a passenger on the wheel is going vertically upwards when the passenger is at point A, 6 metres higher than the centre of the wheel, and is rising.

(b)     The operator of the Ferris wheel stands directly below the centre such that the bottom of the Ferris wheel is level with his eyeline. As he watches the passenger his line of sight makes an angle \(\alpha \) with the horizontal. Find the rate of change of \(\alpha \) at point A.

(a)

\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 3\)     (A1)

\(y = 10\sin \theta \)     A1

\(\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 10\cos \theta \)     M1

\(\frac{{{\text{d}}y}}{{{\text{d}}t}} = \frac{{{\text{d}}y}}{{{\text{d}}\theta }}\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 30\cos \theta \)     M1

at \(y = 6\) , \(\cos \theta  = \frac{8}{{10}}\)     (M1)(A1)

\( \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}t}} = 24\) (metres per minute) (accept \(24.0\))     A1

[7 marks]

(b)     \(\alpha  = \frac{\theta }{2} + \frac{\pi }{4}\)     M1A1

\(\frac{{{\text{d}}\alpha }}{{{\text{d}}t}} = \frac{1}{2}\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 1.5\)     A1

[3 marks]

Total [10 marks]

Many students were unable to get started with this question, and those that did were generally very poor at defining their variables at the start.