Introduction
Calculus can be used to predict sunrise and sunset times around the world. Your task is to use calculus to develop a model that will predict sunrise and sunset times for San Francisco, and compare that model to actual sunrise and sunset times.
Guiding Question
How accurately can you predict sunrise and sunset times in San Francisco and in other cities around the world?
Questions
1. Obtain a chart of sunrise and sunset times for various days of the year in San Francisco.
2. Use this chart to derive a sinusoidal equation for the number of minutes of daylight in a day as a function of the day of the year.
3. Draw a graph of predicted length of daylight versus day. Use your graphing calculator to get the plotting data. On the same axes, plot the actual length of daylight derived from your chart.
4. Discuss how well the sinusoidal model fits the real data. For example, how much daylight does your function predict there will be on the longest day of the year? Shortest day of the year? How does that compare with the actual amount of daylight that day?
5. Explore other questions such as: What is the greatest change in length of daylight from the table? How does this number compare with the derivative of the sinusoidal function at that time of year?
6. Repeat this project for different cities around the world. What conclusions do you draw?
Internet Resources
Selected US and International City Sunrise/Sunset Charts
Global Sunrise/Sunset time charts
Latitude and Longitudes of US and Canadian Cities
Calculus and Math Definitions
Calculus Help
Visualize Derivatives
Understanding Derivatives through Linear Approximation
The Physical Concept of a Derivative
More Information on Derivatives
Derivative in the Encyclopedia
Using Data To Graph A Sine Curve
Ti-83 Graphing Using Data Points Tutorial (PDF)
Activities
Your write-up must include:
1. Objective: Develop and explain the main question you are trying to answer with this experiment.
2. Data: Record the names of the cities along with the data selected from the sunrise/sunset charts.
3. Procedures: Describe what you did in this project and the method you used to find the sinusoidal model (equation).
4. Visuals: Graph the functions you developed in this project, complete with proper labels and captions for each.
5. Calculations: Show work for all calculations done with written explanations of the calculations.
6. Conclusions: Write in complete sentences all the conclusions arrived at.
Jean Nanjo, Teacher, and Jerry Lee, AP student, collaborated to develop this web hunt.
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