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Calculus, 2ADV C3 2024 MET2 18*

A trapezium has the following dimensions.
 

  1. Show that the total area of the trapezium is given by
  2.          (1 mark)

  3. Find the value of which maximises the area of the trapezium below.   (3 marks)

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i.   

ii.   

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i.   


 

ii.   
   
   

 

(

   

   
 

Calculus, 2ADV C3 2024 HSC 31

Two circles have the same centre . The smaller circle has radius 1 cm, while the larger circle has radius cm. The circles enclose a region , which is subtended by an angle at , as shaded.

The area of is cm, where is a constant and .
 

Let cm be the perimeter of .

  1. By finding expressions for the area and perimeter of , show that  .   (3 marks)
  2. Show that if the perimeter, , is minimised, then must be less than 2.   (3 marks)
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a.   

 
 
 
 
 

 

 
   
   
   
   

 
b.   

 
 
 

 

 
   
   
   
   

 

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a.   

 
 
 
 
 

 

 
   
   
   
   
♦ Mean mark (a) 41%.
COMMENT: Sector/arc calculations used in solution are for those who don’t want to remember formulas.

b.   

 
 
 

 

 
   
   
   
   

 

♦♦♦ Mean mark (b) 12%.
 

Calculus, 2ADV C3 2023 HSC 24

A gardener wants to build a rectangular garden of area 50 m² against an existing wall as shown in the diagram. A concrete path of width 1 metre is to be built around the other three sides of the garden.
 

Let and be the dimensions, in metres, of the outer rectangle as shown.

  1. Show that  = .  (1 mark)

  2. Find the value of such that the area of the concrete path is a minimum. Show that your answer gives a minimum area.  (4 marks)

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a.    
 
 
 
 
 

 
b.   

 
   
   

 

 
   

 

 
 
 

 

♦♦ Mean mark (b) 33%.

Calculus, 2ADV C3 2022 HSC 31

A line passes through the point    and meets the axes at    and  , where .
 

  1. Show that  .  (2 marks)

  2. Find the minimum value of the area of triangle .  (4 marks)

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a.  

 
 
 
   
   
   

 


♦♦♦ Mean mark part (a) 17%.
COMMENT: is the expression of a relationship between the intercepts and not the equation of the line.
b.   
   
   

 

 
   
   

 


 

 
   

♦♦ Mean mark part (b) 29%.

Calculus, 2ADV C3 2020 HSC 25

A landscape gardener wants to build a garden in the shape of a rectangle attached to a quarter-circle. Let and be the dimensions of the rectangle in metres, as shown in the diagram.
 


 

The garden bed is required to have an area of 36 m² and to have a perimeter which is as small as possible. Let metres be the perimeter of the garden bed.

  1. Show that  .  (3 marks)

  2. Find the smallest possible perimeter of the garden bed, showing why this is the minimum perimeter.  (4 marks)

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a.   
 
 
 

♦ Mean mark part (a) 47%.

 

 
 
 

 

b.   

 

 

 

 

Calculus, 2ADV C4 2019 HSC 16c

The diagram shows the region  , bounded by the curve  , where  , the -axis and the tangent to the curve at the point  .
 

  1. Show that the tangent to the curve at    meets the -axis at
     
         .  (2 marks)

  2. Using the result of part (i), or otherwise, show that the area of the region    is
     
         .  (2 marks)

  3. Find the exact value of    for which the area of    is a maximum.  (3 marks)

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i.   
 

 

♦♦ Mean mark part (i) 31%.

 

 

 

ii.   

♦♦♦ Mean mark part (ii) 21%.

 


 

`= 1/(r + 1) – 1/2(1 – (r – 1)/r)“
 
 
 
 

 

iii.   
 
   
   
   

 

♦♦ Mean mark part (iii) 23%.

 
 

 

 
 

 

Calculus, 2ADV C4 SM-Bank 12

Let  

A right-angled triangle has vertex at the origin, vertex on the -axis and vertex on the graph of  , as shown. The coordinates of are  
 

 vcaa-2013-meth-10
 

  1. Find the area, , of the triangle in terms of .  (1 mark)

  2. Find the maximum area of triangle and the value of for which the maximum occurs.  (3 marks)

  3. Let be the point on the graph of  on the -axis and let be the point on the graph of  with the -coordinate .Find the area of the region bounded by the graph of  and the line segment .  (2 marks)

     

 

      vcaa-2013-meth-10i

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  2. ²
     
  3. ²
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i.  
   
 

 

ii.   

♦ Mean mark (Vic) 35%.

 

   

²

 

iii.   

♦♦ Mean mark (Vic) 32%.

 

 
 
 

 

vcaa-2013-meth-10ii

 
 
 
  ²

Calculus, 2ADV C3 2016 HSC 14c

A farmer wishes to make a rectangular enclosure of area 720 m². She uses an existing straight boundary as one side of the enclosure. She uses wire fencing for the remaining three sides and also to divide the enclosure into four equal rectangular areas of width m as shown.
 

hsc-2016-14c
 

  1. Show that the total length, m, of the wire fencing is given by
     
          .  (1 mark)

  2. Find the minimum length of wire fencing required, showing why this is the minimum length.  (3 marks)

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i.  
 
 

 

 

 

ii. 
 

 

 

Calculus, 2ADV C3 2004 HSC 10b

2004 10b

 
The diagram shows a triangular piece of land    with dimensions   metres,  metres and    metres, where  .

The owner of the land wants to build a straight fence to divide the land into two pieces of equal area. Let    and    be points on    and    respectively so that    divides the land into two pieces of equal area.

Let   metres, metres and   metres.

  1. Show that  .  (1 mark)

  2. Use the cosine rule in triangle    to show that
     
           (2 marks)

  3. Show that the value of    in the equation in part (ii) is a minimum when
     
         .  (4 marks)

  4. Show that the minimum length of the fence is   metres, where  . 

     

    (You may assume that the value of    given in part (iii) is feasible.)  (2 marks)

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i.    
 

 

 

ii. 

   

 

iii.  
 
   

 

 

 

iv. 

 
 
 
 
 
 
 
 
 

 

Calculus, 2ADV C3 2014 HSC 16c

The diagram shows a window consisting of two sections. The top section is a semicircle of diameter    m. The bottom section is a rectangle of width    m and height    m.

The entire frame of the window, including the piece that separates the two sections, is made using 10 m of thin metal.

The semicircular section is made of coloured glass and the rectangular section is made of clear glass.

Under test conditions the amount of light coming through one square metre of the coloured glass is 1 unit and the amount of light coming through one square metre of the clear glass is 3 units.

The total amount of light coming through the window under test conditions is    units.

  1. Show that  .   (2 marks)

  2. Show that  .    (2 marks)

  3. Find the values of    and    that maximise the amount of light coming through the window under test conditions.   (3 marks)

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i.   

 

 

♦ Mean mark 35%
ii.   
 
   
   

 

 
 
 
 

 

♦ Mean mark 38%
COMMENT: A sanity check for your answer could be to compare your answers to the perimeter restriction of 10m.
iii.  
 
 

 

 

 

 
 

 

Calculus, 2ADV C3 2008 HSC 10b

 

The diagram shows two parallel brick walls    and    joined by a fence from    to  .  The wall    is    metres long and  .  The fence    is    metres long.

A new fence is to be built from    to a point    somewhere on  .  The new fence    will cross the original fence    at  .

Let    metres, where  .

  1. Show that the total area,    square metres, enclosed by    and    is given by
     
         .   (3 marks)

  2. Find the value of    that makes    as small as possible. Justify the fact that this value of    gives the minimum value for  .  (3 marks)

  3. Hence, find the length of    when    is as small as possible.    (1 mark)

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i.

♦♦♦ Low mean marks highlighted (although exact data not available before 2009).

 

 

 
 
 
 
 
 
 

 

ii.   

MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that  and are constants when differentiating. 

 

 

iii.   

 
 

 

Calculus, 2ADV C3 2011 HSC 10b

A farmer is fencing a paddock using    metres of fencing. The paddock is to be in the shape of a sector of a circle with radius    and sector angle  in radians, as shown in the diagram.

2011 10b

  1. Show that the length of fencing required to fence the perimeter of the paddock is
      
       .    (1 mark)

  2. Show that the area of the sector is  .    (1 mark) 

  3. Find the radius of the sector, in terms of  , that will maximise the area of the paddock.    (2 marks)

  4. Find the angle    that gives the maximum area of the paddock.    (1 mark)

  5. Explain why it is only possible to construct a paddock in the shape of a sector if
     
            (2 marks)

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i.   

 
 

 

ii.   .

♦ Mean mark 41%.
TIP: Area of a sector the percentage of the circle that the sector angle accounts for, i.e. radians. .
 
 
 

 

iii. 

♦♦ Mean mark 29%
MARKER’S COMMENT: The second derivative test proved much more successful and easily proven in this part. Make sure you are comfortable choosing between it and the 1st derivative test depending on the required calcs.

 

 

iv.   

♦♦♦ Mean mark 20%
 

 

v. 

♦♦♦ Mean mark 4%. A BEAST!
MARKER’S COMMENT: When asked to ‘explain’, students should support their answer with a mathematical argument.

 

Calculus, 2ADV C3 2012 HSC 16b

The diagram shows a point    on the unit circle    at an angle    from the positive  -axis, where  .

The tangent to the circle at    is perpendicular to  , and intersects the  -axis at  ,  and the line    intersects the  -axis at  .
 

 
 

  1. Show that the equation of the line    is  .    (2 marks)

  2. Find the length of    in terms of  .    (1 mark)

  3. Show that the area,  ,  of the trapezium    is given by 
     
             (2 marks)

  4. Find the angle    that gives the minimum area of the trapezium.    (3 marks)

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i.    

♦♦ Mean mark 20%

 

ii.  

 

 

 

iii. 

 

               

♦♦ Mean mark 24%

 

 
  ²

 

iv.  

 
 
 
 
Mean mark 19%
IMPORTANT: Look for any opportunity to use the identity → often a key to simplifying difficult trig equations.
 

 

 

IMPORTANT: Is the 1st or 2nd derivative test easier here? Examiners have often made one significantly easier than the other.

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