A Hurricane Direction Shift

The movement of a hurricane is modelled by vector h. 

It is moving at a speed of 20 kph, with the direction of 165˚ above the equator, when portrayed on a flat map of the Earth. 

a) Write h in component form.

[2 marks]

The hurricane makes landfall. Its movement is now modelled by vector l, \(\left( \frac{15\sqrt{3}}{2}\mathbf{\text{i}},\mathbf{\ }\frac{15}{2}\mathbf{j} \right)\).

b) Find the amount by which the hurricanes speed has decreased and state the hurricanes new direction.

[3 marks]

Filling a Weather Balloon

The height (h km) that a weather balloon can reach is related to the volume (v m₃) of helium in it at sea level by the equation:

\[h = \frac{8v^{2}}{5} – \frac{32v^{3}}{255}\ ,\ v \leq 12\]

a) Find the volume of helium required to achieve the maximum height and state this height.

[5 marks]

A Solar Sine Curve

You are given the equation

\[f\left( x \right) = 5\cos\theta – 8\sin\theta\]

a) Express f(x) in the form \(R\cos{(\theta + \alpha})\) where \(R > 0\) and\(\ 0 < \alpha < \pi\). Write R in surd form and give the value of α correct to 4 decimal places.

[4 marks]

The temperature of a solar panel, T ˚C, can be modelled by the equation

\[T = 20 + 5\cos\frac{4x}{15} – 8\sin\frac{4x}{15}\ ,\ 0 \leq x \leq 72\]

Where x is the time in hours since 10pm one evening?

b) Calculate the maximum value of T predicted by this model and the value of x, to 2 decimal places, when this value first occurs.

[4 marks]

c) Calculate the times during the first 24 hours when the temperature is predicted, by this model, to be exactly 17 ˚C

[4 marks]

An Advancing Hurricane

A hurricane is heading from the Atlantic Ocean to Miami, Florida. If the centre of the hurricane goes within a 50 kilometre radius of the city centre, the city will sustain very severe damage.

A meteorologist models the city centre as a fixed particle located at the point (240 , 200). They then model the centre of the hurricane as travelling on a path relative to the city defined by \(y = kx + 20\). 

The unit of distance for the meteorologist’s coordinate system is kilometres.


a) Find the values of k such that the city does not sustain severe damage.

[8 marks]

b) State one limitation of the model.

[1 mark]

Separated Satellites

2 weather-monitoring satellites orbit the earth.

One is in a circular orbit C₁, the other orbits in an extreme ellipse, C₂ so that it can get closer to the surface. 

These orbits can be modelled by the equations below:

\[C_{1}:\left( x + 2\sqrt{17} \right)^{2} + y^{2} = 66\]

\[C_{2}:x = 10\cos t,\ y = 4\sqrt{2}\sin t,\ 0 \leq t \leq 2\pi\]

Give the x coordinate of the points of intersection of the curves C₁ and C₂, given that

\(- 5 \leq x \leq 0\). Give an exact answer in the form

\[A\sqrt{1122} + B\sqrt{17}\]

[7 marks]

Increasing Rainfall

As the atmosphere warms, the air holds more water vapour, and this could lead to more intense rainfall events, resulting in an increased flood risk.  In this question, assume every year has 365 days.

The graph shows how the average rainfall (in mm/day) on a rain day at Falmer, Sussex, England has varied with time.

increasing rainfall

The individual points on the graph show the observed average rainfall in mm/day.

a) What was the observed average rainfall  on a rain day in 2000? Give your answer in mm/day.

[1 mark]

b) In an unrealistic model, a student presumes that every day that year was a rain day. Using this information and your answer to (a) find the total amount of rain that fell in that year. Give your answer in metres, to 2 significant figures.

[2 marks]

For the next questions refer to the line of best fit.

c) Calculate the percentage increase in average rainfall between the years 1939 and 1998.

[3 marks]

d) Calculate the percentage increase in the amount of rainfall between the years 1910 and 1920, and the years 2000 and 2010.

[4 Marks]