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Calculus, EXT1 C1 2018 HSC 12b

A ferris wheel has a radius of 20 metres and is rotating at a rate of 1.5 radians per minute. The top of a carriage is metres above the horizontal diameter of the ferris wheel. The angle of elevation of the top of the carriage from the centre of the ferris wheel is .
 


 

  1. Show that  .  (1 mark)

  2. At what speed is the top of the carriage rising when it is 15 metres higher than the horizontal diameter of the ferris wheel? Give your answer correct to one decimal place.  (2 marks)

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Calculus, EXT1 C1 2005 HSC 7a

An oil tanker at    is leaking oil which forms a circular oil slick. An observer is measuring the oil slick from a position  , 450 metres above sea level and 2 kilometres horizontally from the centre of the oil slick.
 


 

  1. At a certain time the observer measures the angle, , subtended by the diameter of the oil slick, to be 0.1 radians. What is the radius, , at this time?  (2 marks)

  2. At this time,  radians per hour. Find the rate at which the radius of the oil slick is growing.  (2 marks)

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Calculus, EXT1 C1 2008 HSC 3c

2008 3c
  
 A race car is travelling on the  -axis from    to    at a constant velocity,  .

A spectator is at    which is directly opposite  , and    metres. When the car is at  , its displacement from    is    metres and  , with  .

  1. Show that  .   (2 marks)

  2. Let    be the maximum value of  .  

     

    Find the value of    in terms of    and  .   (1 mark)

  3. There are two values of    for which  .

     

    Find these two values of  .   (2 marks)

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MARKER’S COMMENT: The most successful students used   and the chain rule to find .
 
 

 

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iii. 

MARKER’S COMMENT: Many students lost marks by not giving their answer in the specified range. Be careful!

 

 

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.

 

 

 

 
 

 

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♦♦♦ Mean mark 14%