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Calculus, EXT1 C1 2004 HSC 3c

A ferry wharf consists of a floating pontoon linked to a jetty by a 4 metre long walkway. Let    metres be the difference in height between the top of the pontoon and the top of the jetty and let    metres be the horizontal distance between the pontoon and the jetty.
 

2004 3c
 

  1. Find an expression for    in terms of  .  (1 mark)

When the top of the pontoon is 1 metre lower than the top of the jetty, the tide is rising at a rate of 0.3 metres per hour.

  1. At what rate is the pontoon moving away from the jetty?  (3 marks)

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Calculus, EXT1 C1 2014 HSC 13b

One end of a rope is attached to a truck and the other end to a weight. The rope passes over a small wheel located at a vertical distance of  40 m above the point where the rope is attached to the truck.

The distance from the truck to the small wheel is , and the horizontal distance between them is  . The rope makes an angle with the horizontal at the point where it is attached to the truck.

The truck moves to the right at a constant speed of   , as shown in the diagram.
 


 

  1. Using Pythagoras’ Theorem, or otherwise, show that  .   (2 marks)

  2.  Show that  .    (1 mark)

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Calculus, EXT1 C1 2010 HSC 2d

A radio transmitter is situated 6 km from a straight road. The closest point on the road to the transmitter is .

A car is travelling away from along the road at a speed of −1.  The distance from the car to is    and from the car to is  .
 

2010 2d
 

Find an expression in terms of    for  , where is time in hours.    (3 marks) 

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Calculus, EXT1 C1 2012 HSC 14c

A plane takes off from a point . It flies due north at a constant angle to the horizontal. An observer is located at , 1 km from , at a bearing 060° from .

Let km be the distance from to the plane and let km be the distance from the observer to the plane. The point is on the ground directly below the plane.
 

2012 14c
 

  1. Show that  .   (3 marks)

  2. The plane is travelling at a constant speed of 360 km/h.
  3. At what rate, in terms of  , is the distance of the plane from the observer changing 5 minutes after take-off?    (2 marks)

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i. 

♦ Mean mark 42%
IMPORTANT: Students should always be looking for opportunities to use the identity to clean up calculations with trig functions.

 

 

 

 
 

 

ii.  

♦♦♦ Mean mark 14%