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A Solar Sine Curve

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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]

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