methods U3/4 Application SAC 2018 Technology free PDF

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VCE Methods SAC
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Name: Mathematical Methods Unit 3 Application Task SAC 2018 Technology Free Component VCAA Mathematical Methods Formula Sheet provided

81 marks

PART 1 (46 marks) Melbourne City Council is proposed a theme park for Carlton Gardens. Within the park there will be a roller coaster. This SAC explores the design of the roller coaster.

The following sketch represents a number of sections of the design for the track, which is going to be constructed in this project. Each labelled section of track can be modelled as a mathematical function.

The roller coaster train starts at the platform, which is 5 metres high above the ground. Assume the ground level is the x-axis and the height of the train above the ground is described by the y-coordinate. The point of origin has been labelled.

Question 1 (Section ) (7 marks) The roller coaster train starts at the platform, which is 5 metres high above the ground, and will climb to a height of 25 metres through the straight track labelled as section in the diagram. When the train reaches its highest point it has a horizontal distance of 16 metres from the starting point. a) State the coordinate of the starting point and the highest point of the slope. 2 marks

y=mx + c , find the positive b) If section of the track can be modelled as a linear function constant m and c, and specify the restricted domain of the function

3 marks c) Sketch the graph of the function found in b). Label the coordinates of the starting point and end point. 2 marks

Question 2 (Section

) (4 marks)

Section refers to a flat track of the roller coaster project. The length of the track is 4 metres. a) Give a function describing the flat track, and specify its domain.

2 marks b) Add a sketch of section

to the graph above, labelling endpoints.

2 marks

Question 3 (Section

) (10 marks)

In section , the train will accelerate and travel down in a very fast manner. Section 2 track can be modelled by a quadratic function in the form a(x )=d ( x+ e ) + f . The commencement of section , (20, 25), can be modelled as the turning point of the quadratic and the section extends from (20, 25) to (25, 18)

a) State values for e and f

2 marks b) Determine the value of d , and state the equation of a(x) and its restricted domain

3 marks c) Sketch a(x) onto the graph within the appropriate domain. Label endpoints. 2 marks

d) Compare a(x) to the graph of y = x 2 . Describe the transformations that take 2 y = x to the equation of a(x) you stated in b)

3 marks

Question 4. (Section

) (12 marks)

Section of the track involves some up and down motion for the train. The designer is unsure of the appropriate function the use. Two models are proposed, one being a trigonometric function the other being a cubic function. The section commences at A(25,18) and ends at C(45,18),as shown in the diagram.

Trigonometric option: Consider the function

k ( x )= p sin n ( x+k ) +m

a) determine values for p, n, k and m

4 marks b) State the domain and range of k(x)

2 marks

Cubic option: Consider the function

h ( x )=

−2 ( x+ p )( x +q )( x +r ) +s , x ∈[ 25,45] 13125

c) If the section commenced at A(25,18) and ended at C(45,18), determine values for p, q, r and s

4 marks d) Add k(x) onto the graph within the appropriate domain, labelling appropriately. 2 marks

Question 5 (Section

) (6 marks)

In the final section, the track is being modelled by the hyperbola

j ( x )=

t +u x −30

a) If the section has end points (45,18) and (60,3), state two equations in terms of t and u

2 marks b) Determine values for t and u

3 marks

c) Determine j(50)

1 mark

Question 6 (7 marks) a) Assist the planner of the track design, express the functions that determine the total roller coaster as the hybrid function T(x). Include equations, restricted domains and either k(x) or h(x) for section .

¿ ¿ ¿ ¿ ¿ T(x)= ¿¿ ¿ ¿ ¿ ¿ ¿

5 marks

b) State the domain and range of T 2 marks

PART 2 (35 marks) Other sections of the track need special consideration. Question 7 ( 7 marks) The planners wanted to use addition of ordinates to improve the variety of designs available. One option was to use πx ) a(x )=2 sin( 10 to form

and

{

−x b ( x )= 2 ,−20 ≤ x ≤ 0 x , 0< x ≤20 2

c (x)=a (x )+ b ( x )

Help the planners decide by sketching a(x), b(x) and then c(x) over the domain x ∈[−20,20 ]

2 + 2 + 3 = 7 marks

Question 8 (12 marks) One planner was keen to include a natural logarithm into the design. She sketched the curve to the right. The endpoints are at (0, 0) and (5, 5) as shown.

She used the general rule d ( x )=a loge (x +b)

a) Write two equations in terms of a and b

2 Marks

b) Show that

d ( x) =

5 loge (x +1) loge 6

2 marks

The planner thought the inverse of d(x) may be a more appropriate track design. c) Will

d−1 (x )

be a function? How do you know?

2 marks d) Determine the equation of

d

−1

(x )

2 marks e) State the domain and range of

d−1 (x )

2 marks f) Sketch the graph of

d−1 ( x ) on the graph including labelled intersection points. 2 marks

Question 9 (16 marks) Composite functions might provide the planners appropriate track designs. Consider the three functions listed below. a) State their domain and ranges.

s (x ) =loge (x+ 5)

t ( x )= √ x−5−5

+5

u ( x ) =e

x−5

6 marks b) Which of the following would exist and state why.

s (t(x ) )

u(s ( x ))

t (u ( x ) )

3 marks The planners eventually decide to use

w ( x )= m(n ( x) ) where

m ( x )=x 2 +1

and

n ( x )= √ x−1

c) State the domain and range of each m(x) and n(x)

4 marks d) State w(x), and its domain and range.

3 marks...