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.
Follow the Activity Procedures:
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