Question 1 (13 Marks)
wall
a. The graph below shows the side view of a section of a skate park that is proposed to be built near an
existing wall. A linear ramp (AB) is constructed from a triangular prism 20 metres deep. The equation
x 1
used to model the linear ramp is y 
 , where y is the height above the ground in metres and x is the
10 5
horizontal distance from the wall in metres.
B
A
C
D
E
i. How far from the wall does the linear ramp (AB) begin?
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1 mark
ii. The point C is vertically below point B. If C is 25 m from A, how high is point B above the ground?
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2 marks
iii.What is the distance from A to B? Express your answer correct to two decimal places.
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2 marks
iv. If concrete weighs 1.8 tonnes per m3, how much does the concrete ramp weigh?
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2 marks
b. The skaters half-pipe (BD) consists of a symmetrical curved timber floor attached to the ramp at B. The
bowl ends at point D. The distance between B and D is 30 metres. The lowest point of the curve is .5 metres
from the ground.
i.
Give the coordinates of point D
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1 mark
ii. Give the coordinates of point E
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1 mark
iii. The equation used to model the bowl is y  a x  b   c . Find the values of a, b and c.
2
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2 marks
iv. Show that y 
2 x 2 56 x 809
is another equation that can be used to model the shape of the bowl.


225 75
50
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2 marks
Question 2 (12 marks)
A roller coaster track is in the shape of a cubic function (side view).
The y-axis represents the height above the ground and the x-axis the horizontal distance, which is at ground
level.
For all questions, only consider the section 0  x  110.
All measurements are in metres (m). Scales are shown in the grid below.
a. If the equation is in the form of y  ax  b  x  c  , use the information in the graph to determine the
values of b and c.
2
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2 marks
b. The roller coaster passes through the point (60, 20). Hence, determine the exact value of a.
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2 marks
c. Determine the starting height of the roller coaster (correct to two decimal places)
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2 marks
d. The engineers decide to move the roller coaster track vertically down by 5 metres to add an underground
tunnel to the roller coaster. Give the new equation of the curve. Write in the form 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑.
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2 marks
e. Sketch the new track in the original grid of previous page labelling the x-intercepts only (correct to 2
decimal places). Give the new domain assuming the ride ends at ground level (y = 0).
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3 marks
f. Determine the horizontal distance-part that is underground. Give correct to 2 decimal places.
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1 mark