Applied/modeling context

4 The equation of a curve is \(y = 2 x - \tan x\), where \(x\) is in radians. Find the coordinates of the stationary points of the curve for which \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).

5. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 10 + 5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight.

1. Show that the height of the water 1 hour after midnight is 12.5 metres.
2. Find, to the nearest minute, the times before midday when the height of the water is 9 metres.

4 The size, \(P\), of a population of a certain species of insect at time \(t\) months is modelled by the following formula. \(P = 5000 - 1000 \cos ( 30 t ) ^ { \circ }\)

1. Write down the maximum size of the population.
2. Write down the difference between the largest and smallest values of \(P\).
3. Without giving any numerical values, describe briefly the behaviour of the population over time.
4. Find the time taken for the population to return to its initial size for the first time.
5. Determine the time on the second occasion when \(P = 4500\). A scientist observes the population over a period of time. He notices that, although the population varies in a way similar to the way predicted by the model, the variations become smaller and smaller over time, and \(P\) converges to 5000 .
6. Suggest a change to the model that will take account of this observation.

3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is \(h\) metres at time \(t\) seconds after the seat is at its lowest point. The height is given by the equation \(h = 15 - 14 \cos ( k t ) ^ { \circ }\), where \(k\) is a positive constant.

1. 1. Write down the greatest height of a seat above the ground.
   2. Write down the least height of a seat above the ground.
2. Given that a seat first returns to its lowest point after 150 seconds, calculate the value of \(k\).
3. Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.

2 The height of the first part of a rollercoaster track is \(h \mathrm {~m}\) at a horizontal distance of \(x \mathrm {~m}\) from the start. A student models this using the equation \(h = 17 + 15 \cos 6 x\), for \(0 \leqslant x \leqslant 40\), using the values of \(h\) given when their calculator is set to work in degrees.

1. Find the height that the student's model predicts when the horizontal distance from the start is 40 m .
2. The student argues that the model predicts that the rollercoaster track will achieve a maximum height of 32 m more than once because the cosine function is periodic. Comment on the validity of the student's argument.

12 The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin ( 30 m - 120 ) ^ { \circ }$$ where \(m\) is the month number, counting January \(= 1\), February \(= 2\), through to December = 12 12

1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month.
   12
2. Using this model, find the month with the highest mean temperature.
   12
3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model. 12
4. 1. State, with a reason, which parameter would be increased because of an overall rise in temperatures.
      [0pt] [1 mark]
      12
5. (ii) State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. \section*{END OF SECTION A}