A Speedy Sea Current

An ocean current separates into 2 different currents at a small island that can be modelled as the origin. Current A heads due south and current B heads on a bearing of 100˚. 

An oceanographer wants to measure the relative surface speeds of these two separate currents, by placing buoys in them off the island and measuring the distance travelled by the buoy in 20 hours.

After 20 hours, the buoy in current A has travelled 150km whilst the buoy in current B has travelled 250km.


a)
 Find the average speeds of the two ocean currents.

[2 marks]

b) Calculate the final distance between the buoys.

[2 marks]

c) Θ is the final bearing of the buoy in current B from the buoy in current A. Show that \(\frac{\sin\theta}{\sin{80}} = 0.93\) to 2 decimal places.

[3 marks]

d) Find the area of the triangle with the island and the two buoys as its
vertices.

[3 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]

A New Renewable

A scientist wishes to develop a new way of generating renewable energy.

They decide to use a large magnet on a large spring, oscillating through a coil of wire, generating a current that could then charge a battery, which in turn would store the energy released from the system.

The system is set into motion with a winch system that can be used to set the spring. The following equation can then be used to describe the oscillations, in metres, t minutes after it has been set into motion:

\[A = 10e^{- 0.05t}\cos{10t}\]

a) On a plot of A against t, show that the t values of the
turning points satisfy the equation \(\tan{10t} = – \frac{1}{200}\)

[4 marks]

The battery can only be charged when the amplitude of the oscillations of the spring are greater than 1m. The amplitudes, A, of the oscillations are found at the turning points of the curve.

b) Give the t value of the last turning point where |A|≥1m, and the battery can no longer be charged.

[4 marks]

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