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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Standard |
The Big Wheel at Fantasy Fun Fayre rotates clockwise at a constant speed completing 15 rotations every hour. The wheel has a diameter of 90 metres and the bottom of the wheel is 6 metres above the ground.
A cabin starts at the bottom of the wheel with the top of the cabin 6m above the ground.
(a) Find the greatest height of the top of the cabin reaches as the wheel rotates.
After \(t\) minutes, the height \(h(t)\) metres above the ground of the top of a cabin is given by the function \(h(t)=51-a\cos bt\).
(b) Find the period of \(h(t)\)
(c) Find the value of \(b\).
(d) Find the value of \(a\).
(e) Sketch the graph of \(h(t)\) , for \(0\le t\le 5\).
(f) In one rotation of the wheel, find the probability that a randomly selected seat is at least 70 metres above the ground. Give your answer to two decimal places.
2. | IB Analysis and Approaches |
The Fun Wheel at the Meller Theme Park rotates at a constant speed.
The height, \(h\) metres, of the point initially at the top of the circumference of the wheel after \(t\) minutes is given by:
$$ h(t) = a \cos(bt) + c $$(a) Find the values of \(a, b \text{ and } c \text{ where } a,b,c \in \mathbb{R} \).
(b) Draw a sketch of the function \(h(t)\) for \(0 < t < 4 \).
3. | IB Analysis and Approaches |
The widest river in the world has a width of 11km at its widest point. Suppose there were a straight length of this river near Awkwardville (A) as shown in the diagram below. Points A and P lie on opposite banks, such that AP is the shortest distance across the river. Point B represents the centre of Bumblingburg which is located on the southern riverbank.
$$PB = 40km, AP = 11km \text{ and angle } A \hat{P}B = 90° $$A boat travels at an average speed of \(12km h^{-1}\).
A bus travels along the straight road
between P and B at an average speed of \(30kmh^{-1}\).
(a) Find the travel time, in hours, from A to B given that the boat is taken from A to P, and the bus from P to B.
(b) Find the travel time, in hours, from A to B given the boat travels directly to B.
There is a point D which lies on the road from P to B. such that \(BD = x km\).
(c) If the boat travels from A to D and the bus travels from D to B, find an expression, in terms of \(x\), for the travel time T, from A to B, passing through D.
(d) Find the value of \(x\) so that T is a minimum.
An excursion involves renting the boat and the bus. The cost to rent the boat is £50 per hour and the cost to rent the bus is £35 per hour.
(e) Find the new value of \(x\) so that the total cost to travel from A to B via D is a minimum.
(f) Write down the minimum total cost for this journey.
4. | A-Level |
The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the following equation:
$$H = k – 8\cos (60t)° + 5\sin (60t)°$$where k is a constant.
(a) Express \(H\) in the form \(H = k - R \cos(60t + a)° \) where \(R\) and \(a\) are constants to be found (\( 0° \lt a \lt 90° \)).
(b) Given that the initial height of the passenger above the ground is 2 metres, find a complete equation for the model.
(c) Hence find the maximum height of the passenger above the ground.
(d) Find the time taken for the passenger to reach the maximum height on the fifth cycle. (Solutions based entirely on graphical or numerical methods are not acceptable.)
(e) It is decided that, to increase profits, the speed of the wheel is to be increased. How would you adapt the equation of the model to reflect this increase in speed?
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