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A new tunnel is built. When there is no toll to use the tunnel, 6000 vehicles use it each day. For each dollar increase in the toll, 500 fewer vehicles use the tunnel.
(i) Find the lowest toll for which no vehicles will use the tunnel. (1 mark)
(ii) For a toll of $5.00, how many vehicles use the tunnel each day and what is the total daily income from tolls? (2 marks)
(iii) If d (dollars) represents the value of the toll, find an equation for the number of vehicles (v) using the tunnel each day in terms of d. (2 marks)
(iv) Anne says ‘A higher toll always means a higher total daily income’.
Show that Anne is incorrect and find the maximum daily income from tolls. (Use a table of values, or a graph, or suitable calculations.) (3 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) 500 less vehicles per $1 toll
12×500=6000
∴$12 toll is the lowest for which no
vehicles will use the tunnel.
(ii) If the toll is $5
5×500=2500 less vehicles
∴ Vehicles using the tunnel
=6000−2500
=3500
∴ Daily toll income =3500×$5
=$17 500
(iii) d = toll
v = Number of vehicles using the tunnel
∴v =6000−500d
(iv) Income from tolls
= Number of vehicles×toll
=(6000−500d)×d
=6000d−500d^2
=500d(12−d)
From the graph, the maximum income from tolls
occurs when the toll is $6.
∴ Anne is incorrect.
Alternate Solution
The table of values shows that income (I) increases
and peaks when the toll hits $6 before decreasing
again as the toll gets more expensive.
∴ Anne is incorrect.
The mass M kg of a baby pig at age x days is given by M = A(1.1)^x where A is a constant. The graph of this equation is shown.
(i) What is the value of A? (1 mark)
(ii) What is the daily growth rate of the pig’s mass? Write your answer as a percentage. (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) When x=0,
1.5 =A(1.1)^0
∴A =1.5 kg
(ii) Daily growth rate
=0.1
=10%
A diver springs upwards from a diving board, then plunges into the water. The diver’s height above the water as it varies with time is modelled by a quadratic function. Graphing software is used to produce the graph of this function.
Explain how the graph could be used to determine how high above the height of the diving board the diver was when he reached the maximum height. (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
We can calculate the height of the board by
finding the y-value at t=0, which is 8 m.
The diver’s maximum height occurs at t=0.5,
which is approximately 9.2 m.
∴ Maximum height above the board
=9.2−8
=1.2 m
Lucy went for a bike ride. She left home at 8 am and arrived back at home at 6 pm. A graph representing her journey is shown.
(i) What was the total distance that she rode during the day? (1 mark)
(ii) How much time did Lucy spend riding her bike during the day? (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) Total distance =35+35
=70 km
(ii) Time away from home=10 hours
Time resting =1+1.5+1
=3.5 hours
∴ Time riding =10−3.5
=6.5 hours
A rectangular playing surface is to be constructed so that the length is 6 metres more than the width.
(i) Give an example of a length and width that would be possible for this playing surface. (1 mark)
(ii) Write an equation for the area (A) of the playing surface in terms of its length (l). (1 mark)
A graph comparing the area of the playing surface to its length is shown.
(iii) Why are lengths of 0 metres to 6 metres impossible? (1 mark)
(iv) What would be the dimensions of the playing surface if it had an area of 135 m²? (2 marks)
Company A constructs playing surfaces.
(v) Draw a graph to represent the cost of using Company A to construct all playing surface sizes up to and including 200 m².
Use the horizontal axis to represent the area and the vertical axis to represent the cost. (2 marks)
(vi) Company B charges a rate of $360 per square metre regardless of size.
Which company would charge less to construct a playing surface with an area of 135 m²? Justify your answer with suitable calculations. (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) One possibility is a length of 10 m, and a width
of 4 m (among many possibilities).
(ii) Length=l m
Width=(l−6) m
∴ A =l(l−6)
(iii) Given the length must be 6m more than the width,
it follows that the length must be greater than 6 m
so that the width is positive.
(iv) From the graph, an area of 135 m² corresponds to
a length of 15 m.
∴ The dimensions would be 15 m×9 m.
(v)
(vi) Company A cost=$50 000
Company B cost =135×360
=$48 600
∴ Company B would charge $1400 less
than Company A.