# Activities

• • • ##### Subject Area

• Math: Algebra II: Systems of Linear Equations and Inequalities
• Math: Geometry: Analytic Geometry

• ##### Author 9-12

45 Minutes

Derive™ 6

## Introduction to GPS 2 - The Three Dimensional Case

#### Activity Overview

In this Derive™ activity, students write the analytic expressions for the four spheres and find the exact point of intersection. They also study the GPS system used to accurately determine your position on the Earth.

#### Before the Activity

• See the attached DFW file for detailed instructions for this activity
• Print pages from the attached DFW file for your class
• #### During the Activity

Distribute the pages to the class.

• Set up a 3D plot window and graph a 3D sphere
• Study the standard equation of a sphere, and examine the conversion equations between rectangular coordinates to spherical coordinates
• Enter the expression for a sphere, plot its graph, and translate it to another point
• Plot the translated sphere and the 3 given equations
• Estimate the point of intersection
• Write the analytic expressions for the four spheres, solve the system, find the exact point of intersection, and verify the estimation

• Study the facts about a GPS system
• Record the spherical coordinates of satellites which transmit signals to GPS
• Note down the time differences recorded by the GPS
• Convert the positions of satellites from spherical to rectangular coordinates
• Enter the expressions for the distances between the GPS and the satellites
• Solve the system and determine the position of the GPS
• Rotate the plot and find the point of intersection
• #### After the Activity

Students answer questions listed on the activity sheet.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary