Notes

Civil engineering applications of Simpson's Rule
Building a highway going through a town
Cross-sectional design of highway - arc length, volume of materials
Design of highway: top X rectangle is asphalt, bottom X rectangle plus cut-out half-circle is concrete, coord system has split down middle
Curve design is a piecewise curve. Given a set of formulas.
Use Simpson's Rule to compute the arc length of the road, and report the total length in ft.
Highway standards require a minimum lane width of 15 feet, 10 feet for side shoulder, 5 feet for inner shoulder, 3 lanes both directions. Total width of X.
Use Simpson's Rule to find the total volume of asphalt, volume of concrete required (use Pappus' Theorem to obtain the volume via V = A * length traveled)

Back Story

You are working as a civil engineer at a construction firm. You have been tasked with helping the company build a brand new highway between Town A to Town B. Your engineering coworker Sally has designed the layout of the road and will provide you with information about it. Your engineering coworker Ralph has designed the cross-sectional layout of the road bed, asphalt, and shoulder. It is your job to estimate the amount of construction materials that will be needed to build the highway. You will use the information provided by Sally to estimate the length of the road, and use the information provided by Ralph to estimate the total volume of concrete and asphalt needed.

Route Layout

Your coworker Sally provides you with a map of curves going through the town, along with a coordinate system and piecewise mathematical functions that represent the curves. This information is provided below.

...

(Explanation of drawing)

Road Design

Ralph provides you with the cross-sectional design of the road. This diagram shows several features of the road design - for example, sloped surfaces to prevent water buildup, and a geometry to allow for expansion and contraction due to heating and cooling.

Estimating the Volume

To estimate the amount of concrete you need, you can use Pappas' Theorem, which states that if the centroid of a cross-sectional shape travels a distance L to form a volume V, then the volume is related to this distance and the cross-sectional area via:

$$ V = A * L $$

where

$A = int_{a}^{b} f(x) - g(x) dx$

is the area between two curves f(x) and g(x) that describe the cross-sectional area of the road bed, and

$L = int_{a}^{b} sqrt{ 1 + \frac{dy}{dx}^2 } dx$

is the arc length (or total distance traveled by the road).

To set up the geometry:

right hand rule

looking forward, at the path traced out by the road

xz plane is the one you are facing, looking out at the road

x plane is across

y plane is out

z plane is up

then the cross-sectional slice of the road bed is on the xz plane

and the arc length is on the xy plane (dy/dx is on the xy plane)

Calc II Questions

Use Simpson's Rule, and the spreadsheet template you used for the prior worksheet, to complete the following exercises. Submit a report that answers each of the following questions with complete sentences.

1) Compute the total length of the road, in feet, using Simpson's Rule and the arc length formula. Justify your choices of N. Why is the centerline of the road used to find the length?

2) Determine the total volume of asphalt and concrete that will be used using Simpson's Rule together with Pappus' Theorem (Volume = Area * Distance Traveled By Centroid)

3) On the morning of the construction project, the engineer who gave you the original map of the site runs up to you, breathlessly, and apologizes, explaining that her map did not show that the road was sloped at a 3% grade the entire length of the road. You don't have your spreadsheets or calculations in front of you - all you have is a pencil, and the back of an envelope. The pressure's on: if you want more material in time for the construction job, you'll have to order it before the dump trucks leave in 15 minutes. How much more concrete and asphalt do you need to order?