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Calculus, 2ADV C3 2019 HSC 15c

The entry points,   and  , to a national park can be reached via two straight access roads. The access roads meet the national park boundaries at right angles. The corner, , of the national park is 8 km from    and 1 km from  . The boundaries of the national park form a right angle at  .

A new straight road is to be built joining these roads and passing through  .

Points    and    on the access roads are to be chosen to minimise the distance,   km, from    to    along the new road.

Let the distance    be    km.
 


 

  1. Show that  .  (3 marks)

  2. Show that    gives the minimum value of  .  (3 marks)

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i.   
 
 

 


 
 
 

 

♦ Mean mark part (i) 41%.

 

 

ii.   
   
   
 
   

 

♦♦ Mean mark part (ii) 31%.


 

Calculus, 2ADV C3 2017 HSC 16a

John’s home is at point and his school is at point . A straight river runs nearby.

The point on the river closest to is point , which is 5 km from .

The point on the river closest to is point , which is 7 km from .

The distance from to is 9 km.

To get some exercise, John cycles from home directly to point on the river, km from , before cycling directly to school at , as shown in the diagram.
 


  

The total distance John cycles from home to school is km.

  1. Show that  .   (1 mark)

  2. Show that if  , then  .   (3 marks)

  3. Find the value of that makes  .   (2 marks)

  4. Explain why this value of gives a minimum for .   (1 mark)

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i.  

 
 

 

ii. 

♦ Mean mark 41%.

 

 

iii. 

♦♦ Mean mark 29%.
♦♦♦ Mean mark 9%.

 

iv.  

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.
i.   
 
 

 

 

ii.   

 
 

 

 
 

 

iii. 

 
 

 

iv.   

 

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.

.

Calculus, 2ADV C3 2013 HSC 14b

Two straight roads meet at    at an angle of 60°.  At time    car    leaves    on one road, and car    is 100km from    on the other road.  Car    travels away from    at a speed of 80 km/h, and car    travels towards    at a speed of 50 km/h.
 

2013 14b

The distance between the cars at time    hours is    km.

  1. Show that  .   (2 marks)

  2. Find the minimum distance between the cars.    (3 marks)

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i.   

♦♦ Mean mark 26%

 
 
   

 

ii.   

♦♦ Mean mark 27%
ALGEBRA TIP: Finding the derivative of (rather than making the subject), makes calculations much easier. ENSURE you apply the test to confirm a minimum.