Project Paper

1
Math 165 – 50 points Name: _________________________________
Project: Modeling Sunrise Time
In this project you will be exploring the sunrise time on the 15th day of each month in a particular year in a
certain location in the United States. You will need to select a location and a year, retrieve data for the
sunrise time at that location, and ultimately find a trigonometric function that models the sunrise time
throughout the year.
Instructions for Submission:
• Include answers and other appropriate responses to each of the following steps 1 – 11.
• Include a complete data table with observed sunrise times, converted sunrise times, predicted
sunrise times, and absolute value difference between converted and predicted sunrise times. You
may refer to the table on page 4 as an example.
• Be sure to note your selected location and year. Remember that you may not use the same
location and year as any other student in the class. (2 points)
• Submit a neat, professional, and organized document. It must be typed using your choice of word
processing software. (3 points)
• Include your scatterplots and other appropriate graphs from your choice of graphing application,
computer algebra software, or statistical software (as applicable). The use of appropriate
technology is expected throughout the project. Hand-drawn graphs are not acceptable.
• Group work is not allowed. While you may get ideas from others, each student must submit
his/her own project.
You will be graded on completion, quality of your model, and quality of your responses. The grading rubric
is available for you to view on Brightspace under Project in the Content section.
Good luck!
Getting Started:
First, you need to select a location (a city or town) in the United States that you are interested in exploring.
You also need to select a particular year – no leap years please!
Note: You may not use the same location and year as any other student in the class.
LOCATION: ______________________________
YEAR: __________
2
Retrieving Your Data:
The United States Naval Observatory (USNO) is a scientific agency with a
primary mission to produce positioning, navigation and timing for the
U.S. Navy and the U.S. Department of Defense. The USNO provides a
wide range of astronomical data and products, and serves as a standard
of time for the entire United States.
Normally, you would use USNO’s website to find the time of sunrise on
the 15th day of each month of the year you selected at the location you
selected; however, the USNO’s website is currently being modernized
and, though it was estimated to become available in fall 2020, the work is
delayed due to COVID-19. Instead, to retrieve data
your data, go to https://www.sunrisesunset.com/USA/.
• Search for your specific U.S. city or town and
click Search USA.
• Find your U.S. city or town in the results and
click Custom Calendar
• Select January for the month
• Enter your desired year and hit Enter on your
keyboard
You will now see a calendar for January of your desired year showing the sunrise/sunset times for the
location you selected. Near the bottom of the page you will see a way to jump through the months.
Record the sunrise time
for the 15th day of each
month.
3
Complete the Following Steps:
1. To account for daylight saving time, add one hour to the observed sunrise times for January,
February, November, and December. (1 point)
2. Convert each sunrise time from #1 to a time in decimal hours. For example, if the sunrise time was
5:17, then you would compute the sunrise time in decimal hours as follows:
17 5 5.28 hours.
60
+ =
Please keep two decimal places and record these times in your data table. (3 points)
3. Create a scatterplot of your data using a calculator such as Desmos (www.desmos.com/calculator).
Use the day of the year as the independent variable and sunrise time (in decimal hours) as the
dependent variable. For example, a sunrise time of 5:17am on August 15th would be the ordered
pair
(227,5.28). Be sure to label both axes appropriately. Include this graph in your project
report. (3 points)
4. Imagine sketching a sine function through your data points on the graph in #3. (12 points)
a) What is the approximate amplitude
A
of the function?
b) What is the approximate vertical shift
D
of the function?
c) What is the approximate period of the function?
d) Now, use the approximated period to calculate the value B. Recall Period
2
B

= .
e) What is the approximate horizontal shift/phase shift of the function?
f) Now, use approximated phase shift (and your approximation of B) to calculate the value C.
Recall Phase Shift
C
B
= .
5. Take your approximations of A, B, C, and D from #4 and create a sine function of the form
f x A Bx C D ( ) sin = − + ( ) . Graph this sine function in the same window (graph) as your
scatterplot. Include this graph with your project report. (2 points)
6. Your sine function from #5 can probably be adjusted to better fit the data points. Tweak your
values of A, B, C, and D to find a sine function of that models the data well when graphed over the
interval
1 365  x . Have patience! It takes take time to find a good model. Clearly state your
final sine function in the form
f x A Bx C D ( ) sin = − + ( ) . Use at least three decimal places for
the value of
B
. Use at least one decimal place for the values of
A, C
, and
D. (10 points)
7. Graph your final sine model from #6 in the same window (graph) as your scatterplot. Include this
graph with your project report. Be sure to label both axes appropriately. (2 points)
8. Discuss thoroughly how well your model fits the data. Be specific. (2 points)
9. Use your model from #6 to predict the sunrise time for the 15th day of each month and record
these times in your data table. Please keep two decimal places. Remember, your calculating
device/program should be in radian mode. (3 points)
4
10. Assess the accuracy of your model by computing the absolute value difference between the
converted sunrise time and the predicted sunrise time for the 15th day of each month. Please keep
two decimal places and record these values in your data table. Also, sum all of the absolute value
differences and report this value as well. (5 points)
11. What is the greatest percent error between converted sunrise time and predicted sunrise time?
Please keep two decimal places. In which month did it occur? (2 points)
Hint:
true value estimated value percent error 100
true value
−
= 
LOCATION: ______________________________ YEAR: __________
Date
Day of
the
Year
Observed
Sunrise Time
(in HH:MM)
Daylight
Saving Time
Adjustment
(in HH:MM)
Converted
Sunrise Time
(in decimal
hours)
Predicted
Sunrise Time
(in decimal
hours)
Absolute Value
Difference Between
Converted and
Predicted Sunrise
Time
January 15th 15
February 15th 46
March 15th 74
April 15th 105
May 15th 135
June 15th 166
July 15th 196
August 15th 227
September 15th 258
October 15th 288
November 15th 319
December 15th 349
SUM =