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Calculus, MET2 2019 VCAA 2

An amusement park is planning to build a zip-line above a hill on its property.

The hill is modelled by  , where    is the horizontal distance, in metres, from an origin and    is the height, in metres, above this origin, as shown in the graph below.
  

  1. Find  .   (1 mark)

  2. State the set of values for which the gradient of the hill is strictly decreasing.   (1 mark)

The cable for the zip-line is connected to a pole at the origin at a height of 10 m and is straight for  , where  . The straight section joins the curved section at  . The cable is then exactly 3 m vertically above the hill from  , as shown in the graph below.

  1. State the rule, in terms of , for the height of the cable above the horizontal axis for  .   (1 mark)

  2. Find the values of for which the gradient of the cable is equal to the average gradient of the hill for  .   (3 marks)

The gradients of the straight and curved sections of the cable approach the same value at  , so there is a continuous and smooth join at .

    1. State the gradient of the cable at , in terms of .   (1 mark)

    2. Find the coordinates of , with each value correct to two decimal places.   (3 marks)

    3. Find the value of the gradient at , correct to one decimal place.   (1 mark)

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a.   
 

b.   


 

c.   


 

d.   


 

e.i.   


 

e.ii.   


 

 

 

e.iii.   
   

Calculus, MET2 2009 VCAA 2

VCAA 2009 2a

A train is travelling at a constant speed of km/h along a straight level track from towards

The train will travel along a section of track

Section passes along a bridge over a valley.

Section passes through a tunnel in a mountain.

Section is 6.2 km long.

From to , the curve of the valley and the mountain, directly below and above the train track, is modelled by the graph of
 

where and are real numbers.
 

All measurements are in kilometres.

  1. The curve defined from to passes through . The gradient of the curve at is – 0.06 and the curve has a turning point at  .
  2.  i. From this information write down three simultaneous equations in , and .   (3 marks)

  3. ii. Hence show that  , and .   (2 marks)

  4. Find, giving exact values
  5.   i. the coordinates of .   (2 marks)

  6.  ii. the length of the tunnel.   (1 mark)

  7. iii. the maximum depth of the valley below the train track.   (1 mark)

The driver sees a large rock on the track at a point , 6.2 km from . The driver puts on the brakes at the instant that the front of the train comes out of the tunnel at .

From its initial speed of km/h, the train slows down from point so that its speed km/h is given by

,

where km is the distance of the front of the train from and is a real constant.

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

  2. Find the exact distance from the front of the train to the large rock when the train finally stops.   (2 marks)

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a.i.   

a.ii.   

a.iii. 

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 a.i.  


 


 


 

  ii.   

♦ Mean mark part (a)(ii) 41%.

 


 


 

b.i.  

   

 

 

  ii.  

♦ Mean mark part (a) 41%.
 
 

 

   iii. 

♦ Mean mark part (b) 50%.


 

c. 

♦ Mean mark part (c) 35%.

 

 

 

d.  

♦ Mean mark part (d) 40%.

 

e.  

♦♦ Mean mark part (e) 32%.

 

Calculus, MET2 2013 VCAA 3

Tasmania Jones is in Switzerland. He is working as a construction engineer and he is developing a thrilling train ride in the mountains. He chooses a region of a mountain landscape, the cross-section of which is shown in the diagram below.

VCAA 2013 2a

The cross-section of the mountain and the valley shown in the diagram (including a lake bed) is modelled by the function with rule

Tasmania knows that    is the highest point on the mountain and that and are the points at the edge of the lake, situated in the valley. All distances are measured in kilometres.

  1. Find the coordinates of , the deepest point in the lake.   (3 marks)

Tasmania’s train ride is made by constructing a straight railway line from the top of the mountain, , to the edge of the lake, . The section of the railway line from to passes through a tunnel in the mountain.

  1. Write down the equation of the line that passes through and    (2 marks)

  2.  i. Show that the -coordinate of , the end point of the tunnel, is    (1 mark)

  3. ii. Find the length of the tunnel    (2 marks)

In order to ensure that the section of the railway line from to remains stable, Tasmania constructs vertical columns from the lake bed to the railway line. The column is the longest of all possible columns. (Refer to the diagram above.)

  1.  i. Find the -coordinate of    (2 marks)

  2. ii. Find the length of the column in metres, correct to the nearest metre.   (2 marks)

Tasmania’s train travels down the railway line from to . The speed, in km/h, of the train as it moves down the railway line is described by the function.

where is the -coordinate of a point on the front of the train as it moves down the railway line, and and are positive real constants.

The train begins its journey at . It increases its speed as it travels down the railway line.

The train then slows to a stop at , that is  

  1. Find in terms of    (1 mark)

  2. Find the value of for which the speed, , is a maximum.   (2 marks)

Tasmania is able to change the value of on any particular day. As changes, the relationship between and remains the same.

  1. If, on one particular day, , find the maximum speed of the train, correct to one decimal place.   (2 marks)

  2. If, on another day, the maximum value of is 120, find the value of    (2 marks)

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  3.  i.
  4. ii.
  5.  i.
  6. ii.
Show Worked Solution

a.  


 

b.   
   

 

 
c.i. 


 

 


 

c.ii.  

 

 

 

d.i. 

♦♦♦ Mean mark part (d)(i) 24%.
MARKER’S COMMENT: Many students unfamiliar with this type of question.
 
 

 


 

♦♦♦ Mean mark part (d)(ii) 22%.
d.ii.   
   

 

e.   
 
 

 

f.  


 

g. 

♦ Mean mark 43%.

 

 

h.  

♦ Mean mark 38%.