Module 6 Project

Applications of Sinusoidal Functions

 

Project Overview

 

Purpose

 

In this project you will develop and apply sinusoidal functions that model the number of hours of daylight for different locations in the world.

 

Process

 

Toward the end of each lesson, under Project Connection, you may be prompted to complete a part of the Module 6 Project.

 

Make sure to save all work from each Project Connection in your course folder. You will submit your Module 6 Project to your teacher at the end of Module 6. Check with your teacher about how you should submit your project work before you begin the project.

 

Presentation

 

The most important part of your project presentation is clearly explaining and supporting your reasoning with relevant visuals and calculations. Make sure to show all your work.

 

Review the Module 6 Project Rubric to ensure you understand how you will be assessed on this project.

 

Project Introduction

 

In this project you will develop equations and apply sinusoidal functions that model the number of hours of daylight for locations in the world at different latitudes. To study the relationship between latitude and the number of hours of daylight, data will be collected and analyzed. Using the example of the number of daylight hours, you will then investigate other sinusoidal phenomena.

 

Part 1

1. Latitude is measured from the equator, with positive values going north and negative values going south. Search the Internet for the latitude of your home location. You may want to use key words like “your city name + latitude” to help your search. Record the information that you find.
   
   

   Home Location	Latitude (rounded to the nearest ten)

   
   
   

   Dorling Kindersley RF/Thinkstock

2.  

   1. Search the Internet to determine the number of hours of daylight at different times of the year for each of the following locations. Using key words like “hours of daylight + latitude” may be helpful.
      • equator (0°)
      • latitude of your home location
      • latitude of 30°
      • North Pole (You may have to search this one separately.)

       

      Record the data in a chart like the following. Note: Times may be in decimal form, so a time of 18:24 (6:24 p.m.) is shown as 18.40 h.

      
      

      Number of Daylight Hours
      	Day Number	Number of Daylight Hours at Equator	Number of Daylight Hours at Latitude of Home Location	Number of Daylight Hours at Latitude 30°	Number of Daylight Hours at North Pole
      January 1	1
      February 1	32
      March 1	61

   2. Use the chart to predict which locations have sinusoidal data.
   3. Is the data periodic? How do you know?
3.  
   1. Create a scatter plot for each location that you researched. You may use Graph Template or a spreadsheet/graphing program of your choice. Label each graph and use an appropriate scale for each of the axes.
   2. Which graph(s) follow a sinusoidal pattern? Explain your reasoning.
   3. Draw a curve of best fit for the graphs that follow a sinusoidal pattern.

Part 2

1.  
   1. Use a sinusoidal regression to find an equation, in the form that represents the hours of daylight, h, as a function of day number, n, for your home location.
   2. State the values of a, b, c, and d to the nearest thousandth.
   3. Use mathematical terms, such as amplitude, period, vertical displacement, and phase shift, to describe how the graph of y = sin x has been transformed to obtain the graph of the function that you found for your home location.
   4. State the range and period.
2.  
   1. Use a sinusoidal regression to find an equation, in the form that represents the hours of daylight, E, as a function of day number, n, for a location at latitude 30°.
   2. State the values of a, b, c, and d to the nearest thousandth.
   3. Use mathematical terms, such as amplitude, period, vertical displacement, and phase shift, to describe how the graph of y = sin x has been transformed to obtain the graph of the function that you found for a location at latitude 30°.
   4. State the range and period.
3. Determine the number of hours and minutes of daylight in the following locations on May 23 (day 144).
   1. your home location
      
   2. a location at a latitude of 30°
4. Explain how the latitude of a location is related to the hours of daylight, and explain how this relationship is illustrated by the differences in the parameters in the two equations.
5. One factor that affects a region’s growing season is hours of daylight. The growing season generally starts when there are 15 or more hours of daylight per day. If hours of daylight is the only factor, what is the predicted start date and end date of the growing season in your home location? Explain the method used to determine these dates.

Part 3

 

Many phenomena exhibit periodic patterns. Find a set of data, other than number of daylight hours, that appears to be periodic. (Some suggestions are tidal data, predator–prey populations, and distance of the moon from Earth.)

1. Use the set of data you selected to sketch a graph that represents the periodic relationship.
2. Draw a curve of best fit for the data.
3. Use an appropriate regression model to determine an equation that will represent the data.
4. If asked to use your regression model to make predictions for your data, how accurate do you expect your predictions will be? Explain.

Save your work in your course folder. You will submit your completed project to your teacher for assessment.

 

Project Assessment

 

Your Module 6 Project will be evaluated by your teacher using the evaluation guidelines in the project rubric. Read the rubric carefully. Make sure you are aware of how you will be assessed. You can print or save a digital copy of the Module 6 Project Rubric as a guide to help you complete your project.

 

Don’t forget to submit your completed Module 6 Project to your teacher at the end of Module 6.