TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

In determining step length, the thought behind the process is that the length is going to change as the slope of the hill changes. The step length may therefore be calculated based on the derivative of the function at different points along the path. The student should be able to determine the point at which steps are to start and from this point determine the width of each individual step.

There should be some discussion of the meaning of the term "length of tread", referring to the run of the step. This should not (but may) be confused with the width of the path.

Prerequisites

The first part of the problem, step lengths, may be completed with a knowledge of derivatives. The section concerning the length of the path which must be paved requires the use of integration to determine the arc length.

Time allotment - time management

Students, working in groups, should be able to complete the work in two hours. The first hour will be more of a discussion of what is necessary to complete the problem and the second will be used to solve it.

Expectations

Future payoffs

This problem relates the use of the derivative to a practical problem involving changes in the slope of landforms.