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Modeling hours of daylight

Desmos has a nice student activity (on teacher.desmos.com) modeling the number of hours of daylight in Florida versus Alaska – which both produce a nice sine curve when plotted on a graph.  So let’s see if this relationship also holds between Phuket and Manchester.

First we can find the daylight hours from this site, making sure to convert the times given to decimals of hours.

Phuket

Phuket has the following distribution of hours of daylight (taking the reading from the first of each month and setting 1 as January)

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Manchester 

Manchester has much greater variation and is as follows:

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Therefore when we plot them together (Phuket in green and Manchester in blue) we get the following 2 curves:

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We can see that these very closely fit sine curves, indeed we can see that the following regression lines fit the curves very closely:

Manchester:

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Phuket:

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For Manchester I needed to set the value of b (see what happens if you don’t do this!) Because we are working with Sine graphs, the value of d will give the equation of the axis of symmetry of the graph, which will also be the average hours of daylight over the year.  We can see therefore that even though there is a huge variation between the hours of daylight in the 2 places, they both get on average the same amount of daylight across the year (12.3 hours versus 12.1 hours).

Further investigation:

Does the relationship still hold when looking at hours of sunshine rather than daylight?  How many years would we expect our model be accurate for?  It’s possible to investigate the use of sine waves to model a large amount of natural phenomena such as tide heights and musical notes – so it’s also possible to investigate in this direction as well.