How to Graph Tides Using Sine Curves.

3758 Words May 11th, 2012 16 Pages
Introduction:

Daylight hours are dependent on sunrise and sunset times for each day which are dependent on seasonal change. The number of daylight hours can be represented by a periodic function. This periodic function can help Alaskan Council predict daylight hours for tourist travelling to watch their Bore Tide. A Bore Tide is a tidal phenomenon in which the leading edge of the incoming tide forms a wave (or waves) of water that travels up a river or narrow bay against the direction of the river or bay's current. Anchorage, Alaska is known for its famous Bore Tides. These bore tides occur at least once a day during high and low tide. Yet, health warnings are applied to viewing bore tides during low tide. Tourists have died by getting
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(Refer to Appendix 5 for the graph of the function:L=417 sin⁡[2π/365 (t-81)]+744). As a regression analysis the function itself was used to find daylight hours. The hours were then compared to the original data points. This comparison resulted in a percentage error being recorded. (Refer to Appendix 5 for the results of the function and percentage errors). The highest percentage error recorded was 7.35% on February 1st 2011. This error could be because February has 29 days in a leap year but 2011 data was used. The error could also occur because only 24 data points were used, if more data points were used a more precise graph would have been generated leaving less space for error.
Question 1e:
Simple deriving and calculus was used to verify that the equation L=417 sin⁡[2π/365 (t-81)]+744 for when the longest and shortest day of the year occur. The derivative of the function was started by using the chain rule(dy/dx=dy/du*du/dx). The derivative was found to be: dy/dx=2π/365 417 cos⁡(2π/365 t-162π/365). dy/dx was made to equal 0 therefore the equation will end up looking like: 0=cos⁡(2π/365 t-162π/365). When dealing with a cos curve on a unit circle cos can only be 0 at 90 degrees (π/2) and 270 degrees (3π/2) therefore meaning that to find the longest and shortest days (2π/365 t-162π/365) is made
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