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Trigonometry, 2ADV T3 SM-Bank 16

Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris wheel at point . The height of above the ground, , is modelled by  , where is the time in minutes after Sammy enters the capsule and is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.
 


 

  1. State the minimum and maximum heights of above the ground.  (1 mark)

  2. For how much time is Sammy in the capsule?  (1 mark)

  3. Find the rate of change of with respect to and, hence, state the value of at which the rate of change of is at its maximum.  (2 marks)

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Calculus, 2ADV C3 SM-Bank 13

The figure shown represents a wire frame where is a convex quadrilateral. The point is on line segment with   and  , where is a positive constant.

Let    where  
 

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  1. Find and in terms of and .  (2 marks)

  2. Find the length, cm, of the wire in the frame, including length , in terms of and .  (1 mark)

  3. Find  ,  and hence show that  when  .  (2 marks)

  4. Find the maximum value of if  .  (1 mark)

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iii.   

♦ Mean mark (Vic) 35%.

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iv. 

♦♦♦ Mean mark (Vic) 5%.

 

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Calculus, 2ADV C3 2007 HSC 10b

The noise level, , at a distance metres from a single sound source of loudness is given by the formula

Two sound sources, of loudness and are placed metres apart.
 

The point lies on the line between the sound sources and is metres from the sound source with loudness

  1. Write down a formula for the sum of the noise levels at in terms of .  (1 mark)

  2. There is a point on the line between the sound sources where the sum of the noise levels is a minimum.

     

    Find an expression for in terms of , and if is chosen to be this point.  (4 marks)

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Calculus, 2ADV C3 2009 HSC 9b

An oil rig,  ,  is 3 km offshore. A power station,  ,  is on the shore. A cable is to be laid from  to  .  It costs $1000 per kilometre to lay the cable along the shore and $2600 per kilometre to lay the cable underwater from the shore to  .

The point    is the point on the shore closest to  ,  and the distance    is 5 km.

The point    is on the shore, at a distance of    km from  ,  as shown in the diagram.


  

  1. Find the total cost of laying the cable in a straight line from    to    and then in a straight line from    to  .   (1 mark)

  2. Find the cost of laying the cable in a straight line from    to  .  (1 mark)

  3. Let    be the total cost of laying the cable in a straight line from    to  ,  and then in a straight line from  to  .
     
    Show that  .    (2 marks)

  4. Find the minimum cost of laying the cable.    (4 marks)

  5. New technology means that the cost of laying the cable underwater can be reduced to $1100 per kilometre.

     

    Determine the path for laying the cable in order to minimise the cost in this case.    (2 marks)

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♦♦ Although specific data is unavailable for question parts, mean marks were 35% for Q9 in total.
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IMPORTANT: Tougher derivative questions often require students to deal with multiple algebraic constants. See Worked Solutions in part (iv).

 
MARKER’S COMMENT: Check the nature of the critical points in these type of questions. If using the first derivative test, make sure some actual values are substituted in.

 
   

 

 

v.   

 
 

 

 

 

MARKER’S COMMENT: Many students failed to interpret a correct calculation of    as providing no solution.

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