## Modelling Sunrise and Sunset Times

The graphs of the sunrise and sunset times in London (U.K.) throughout the year are shown below:

Can you explain the kinks in the curve?

Accurate data for various locations can be found at http://www.timeanddate.com/worldclock/sunrise.html

By making a simple adjustment, the sunrise and sunset times for London (U.K.) (without kinks !!!) can be modelled using the sine function.

Consider the sine function: $f(x)={a}\sin(bx+c)+d$

The project is to choose the values of the coefficients a, b, c and d to best model the annual cycle of sunrise (or sunset) times in London (U.K.).

Hints:

• On what dates are the vernal equinox and autumnal equinox?
• On what dates are summer solstice and winter solstice?
• Consider how the graph changes for Peterborough (U.K.) a city 75 miles North of London (U.K.)
• Find the functions that best models the sunrise and sunset times at Peterborough (UK).
• Consider how the graph changes for Bristol (U.K.), a city 107 miles West of London (U.K.)
• By looking at the data, test your predictions.
• Investigate how change of latitude alters the graph of the sunrise and sunset times.
• Investigate how change of longitude alters the graph of the sunrise and sunset times.

This is one of the many Enrichment tasks on this site.