STATEMENT OF PROBLEM

A straight four-foot wide path going up a hill must be paved to a point and then continue with steps. The paving can only be done on portions of the path that have a slope of magnitude less than 1/3. 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.

Input := 

PathData = {{0,50},{20,47},{40,43},{60,38},{80,32},
	{100,25},{120,17},{140,8},{160,6},{180,2},{200,0}};

Input := 

TableForm[
	PathData,
	TableHeadings ->
		{None,{"Dist from top of hill [feet]",
						"hill height [feet]"}},
	TableAlignments -> Center]

Output =

   Dist from top of hill [feet]   hill height [feet]
                0                         50

                20                        47

                40                        43

                60                        38

                80                        32

               100                        25

               120                        17

               140                        8

               160                        6

               180                        2

               200                        0

Input := 

p1 = ListPlot[PathData,
	PlotStyle -> PointSize[.02]]

Output =

-Graphics-

All relationships are in feet. For example, the ordered pair (60,38) represents a horizontal distance of 60 feet from the top of the hill, and 38 vertical feet from the bottom.

1) Fit the hill side data to a 3rd degree polynomial of the form
y = A x^3 + B x^2 + C x + D.
Does this offer a reasonable fit?
You'll use this path equation to answer the rest of the questions.

2) How many steps are necessary, and what are their individual lengths, i.e. tread lengths? (You'll have to build a table since each step will likely have a unique tread length.)

3) How long is the paved section of the path? (Be sure to give the true length -- not the horizontal length.)

4) The cost of paving is $2.00 per square foot and the cost of putting in steps is $2.50 per square foot (horizontal area only). How much will the path cost to build?

STATEMENT OF PROBLEM

1) Fit the hill side data to a 3rd degree polynomial of the form y = A x^3 + B x^2 + C x + D. Does this offer a reasonable fit? You'll use this path equation to answer the rest of the questions.

2) How many steps are necessary, and what are their individual lengths, i.e. tread lengths? (You'll have to build a table since each step will likely have a unique tread length.)

3) How long is the paved section of the path? (Be sure to give the true length -- not the horizontal length.)

4) The cost of paving is $2.00 per square foot and the cost of putting in steps is $2.50 per square foot (horizontal area only). How much will the path cost to build?

1) Fit the hill side data to a 3rd degree polynomial of the form
y = A x^3 + B x^2 + C x + D.
Does this offer a reasonable fit?
You'll use this path equation to answer the rest of the questions.