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Formula for a sine curve given the amplitude and either the period, frequency or angular frequency.

(i) In london in 2002, the maximum number of daylight hours in a day was 16.63, and this was recorded in week 25.The minimum number of daylight hours in a day was 7.82, and this was recorded in week 51. The number of daylight hours in a day can be modelled approximately by using a sine function.

Use the information given above to find the following values, showing details of your calculations.

(a) the amplitude of the function,

(b) the period (in weeks) of the function,

(c) the vertical displacement of the curve

(d) the phase shifdt of the curve

(ii)
(a) Using your answers to part (i), write down the equation of the curve, defining your variables clearly.

(b) Use this model to determine approximately, how much daylight there was per day in London in week 13 of 2002.

Solution Preview

I would begin this problem by doing a rough sketch (graph) of the information given. This will help a lot with visualizing what they're asking.
a) Amplitude gives a measure of the vertical variation of the function, i.e. the maximum deviation from the average value. To get amplitude, find the difference between the max value and the min value (in this case, 16.63 and 7.82 hours respectively) and divide by 2.
So A = (16.63 - 7.82)/2

b) Remember that a function that is sinusiodal in time has a period that can be measured by finding the time for one full "cycle", i.e. by measuring the distance from peak to peak (or trough to trough). here they have given you the times at which the function is a maximum and when it is a minimum. So you can figure out the distance from the first peak to the first trough (i.e. 51-25 weeks). Since this represents only half the cycle, you have to extrapolate to get the length of the full cycle. The period, which is one full cycle, is therefore twice the time difference between ...

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