This exercise uses the notions of arc length and slope to determine the cost of building a path up a slope that has steps when it's steep and is paved otherwise. A straight four-foot wide path going up a hill must be paved to a point and then continue with steps. The rest of the hill must be furnished with steps. Each step will rise exactly 8 inches high. Surveyors have gathered the following elevation data along the path.
All relationships are in feet.
You'll use this path equation to answer the rest of the questions. 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. The first part of the problem, step lengths, may be completed with a knowledge of derivatives.