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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: __________

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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.

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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)

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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 =