AP Calculus AB / Mr. Hansen
5/3/2005

Name: _________________________

Worksheet in Preparation for Group Projects

 

Instructions: You may work on this essay question at home or in the library. Hand it in no later than 10:45 a.m. on Wednesday, May 4.

 

 

 

One possible group project that we discussed was the task of analyzing the mathematics involved in ballistic missile defense. Specifically, we would like to know exactly where to aim an antimissile missile, and exactly how much thrust to give it, in order to intercept and destroy an incoming enemy missile. This problem is extraordinarily difficult, and the U.S. government plans to spend many billions of dollars on it in the years ahead.

 

 

 

Begin by assuming that we have a table of n ordered sextuples. (If you want to be a little less general, let n = 10 so that you have something concrete to work with.) The 6 data elements of each ordered sextuple are as follows:

1, 2, and 3: enemy missile coordinates
4: 95% confidence radius (in meters)
5: date on which measurement was taken
6: time (to nearest millisecond) at which measurement was taken

Because it is impossible to measure position precisely, even for a stationary observer using a GPS receiver on the ground, item #4 is an estimate of the radius of an imaginary ball about the given coordinates 1, 2, and 3 such that we are 95% certain that the missile is within that ball.

 

 

1.

The first question to address is whether 1, 2, and 3 should be measured in conventional xyz coordinates or in some other coordinate system. Please write a paragraph. You may wish to research this issue in a book or on the Web.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

Next, describe a process whereby techniques used in our class can be used to predict the next few ordered sextuples (n + 1, n + 2, etc.). Obviously, this will consist of several paragraphs of mathematical speculation for now, since doing it for real would be a real project. Hint: Consider Euler’s Method. Also address this question: What happens to data item 4 (the 95% confidence radius) as we go beyond the nth ordered sextuple?