Anna is sitting in a carriage of a Ferris wheel which is revolving. The height, , in metres above the ground of the top of her carriage is given by

,

where is the time in seconds after Anna's carriage first reaches the bottom of its revolution and and are constants.
 

The top of each carriage reaches a greatest height of 39 metres and a smallest height of 3 metres.

1. Find the value of and .   (2 marks)

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2. How many seconds does it take for one complete revolution of the Ferris wheel?   (1 mark)

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3. Billie is in another carriage. The height, , in metres above the ground of the top of her carriage is given by

,

1. where and are as found in part (a).
2. During each revolution, there are two occasions when Anna's and Billie's carriages are at the same heights. At what two heights does this occur? Give your answer correct to 2 decimal places.    (4 marks)

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Find the coordinates of the points of intersection of the graph of the relation

  with the line  , for    (3 marks)

The graphs of  ,  where is a real constant, have a point of intersection at  

1. Find the value of .  (2 marks)
   

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2. Find the  -coordinate of the other point of intersection of the two graphs, given    (1 mark)
   

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The UV index, , for a summer day in Newcastle East is illustrated in the graph below, where    is the number of hours after 6 am.
 

The graph is most likely to be the graph of