Read This Before You Begin

This worksheet will utilize your experience last worksheet implementing Simpson's Rule in a spreadsheet. You will be applying this spreadsheet to estimate the volume of concrete and asphalt needed to build a road.

Student expectations

You will submit a one-page report that answers each of the questions.

You will need to do some analytical work, and some numerical work. You will need to show your work for both kinds of questions.

If a question asks for a plot or a chart, attach it to your report and refer to it in your answer.

Answer questions with complete sentences.

Problem setup

(Essential information on handout, and go through with slides)

Back Story and Background: Highway Curves

Lake Stinky story
Building detour highway between two towns
We have been tasked with ordering the materials required to build the highway
We have to figure out the volume of concrete and volume of asphalt to order

Problem Procedure:

First, we want to be able to describe the road with math
Then we want to know the length of the road
Then we can estimate the volume of the road
That will allow us to calculate the volume of materials required to build it

Information from Sally: Coordinate System

information provided by colleague to describe the road quantitatively
circular part, two linear parts
make up our own coordinate system

Calculation: Road Length

Arc length calculation, review arc length formula
Two lines, one circle - how to turn a circle into a function?

Information from Naveen: Road Cross-Section

Cross sectional design of highway, arc length, volume of materials - highway standards require minimum lane width of 15 feet, 10 feet for side shoulder, 5 feet for inner shoulder.
Our new highway has 4 lanes, total width of 4 * 15 feet + 2 * 10 feet = 60 + 20 = 80 feet

Calculation: Areas and Volumes

Pappus' Theorem: center of gravity, V = A L

Explain that this is a multi-step problem, will require critical thinking skills, work up front, articulating your answer

Worksheet Contents

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 and Town B. Recently a giant sinkhole several miles wide opened up and flooded, creating a giant lake called Lake Stinky, that prevents a highway from directly connecting the two towns. Your job is to estimate the volume of materials required to build the road.

Your engineering coworker Sally has decided on the route the road will take, and will provide you with information about where the road goes. Your engineering coworker Naveen 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 Naveen to estimate the total volume of concrete and asphalt needed.

Background: Highway Curves

Some information about the design of highway curves: point of curve (PC), point of tangency (PT), circular curve connecting the two points. Many factors affecting decision - most important is speed of curve, which determines radius of curve.

During the initial survey work, evenly-spaced points are staked out along the proposed path of the highway. If two straight segments must be connected by a curve, a circle is drawn whose tangent line at the start of the curve, point PC, is the segment of road leading up to the curve. The road follows the arc of this circular curve until the tangent line to the circle is parallel to the segment of road leading away from the curve. This point is the point of tangency (PT) and is where the curve transitions back to a straight section.

In addition to the problem of designing the correct route for the road, the road must be designed to be safe and last a long time. For this reason, highway curves are typically sloped: both to offset the centripetal forces created by a vehicle going around a curve, and to prevent buildup of rainwater or debris. This means the road bed design will be different for a curved section than for a straight section.

Problem Procedure

Talk about how we'll solve this problem:

To estimate how much concrete and asphalt to order, we need to know the volume.

To find a volume, we need an area and a length.

Sally is giving us a coordinate system and mathematical description of the road location. We can use that to obtain an arc length.

Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.

Information from Sally: Road Coordinate System

Let's talk about the information that your engineering coworker Sally has provided to you.

There are two long, straight roads that connect to the point of curve (PC) and point of tangency (PT). The highway from Town A to the point PC points precisely Northeast/Southwest and extends for 14 miles, while the highway from the point PT to Town B points precisely Northwest/Southeast and extends for 10 miles. The curve follows the arc of a circle with a radius of 2 miles.

Calculation: Road Length

To calculate the length of the road, which is described by the function $f(x)$ , use the arc length formula:

$L=\int _{a}^{b}{\sqrt {1+(f'(x))^{2}}}dx$

This will require you to create a coordinate system and describe the curve and two straight sections as a function $f(x)$ .

Information from Naveen: Cross-Section of Road

The cross-sectional design of the road must account for elements like sloped surfaces to prevent water buildup, and the ability to expand or contract during heating and cooling.

(Cross-section of road bed on straight sections)

(Cross-section of road bed on curved sections)

(Explanation of drawing)

Calculation: Area and Volume

Given a mathematical description of the cross-section of the road bed, provided by your co-worker Naveen, you can find the cross-sectional area of the road bed using:

$A=\int _{a}^{b}(f(x)-g(x))dx$

To calculate the area of concrete, you can find the cross-sectional area of the concrete portions of the road bed.

To calculate the area of asphalt, you can find the cross-sectional area of the asphalt portions of the road bed.

Pappus' Centroid Theorem allows you to calculate the volume formed by rotating an area over a certain arc length. You can find the volume by multiplying the area times the distance traveled by the centroid of the area. This gives the formula for the volume:

$V=AL$

Your Job

Your job in the construction process is to estimate the total volume of concrete or asphalt you will need to order to construct the highway. You can use the mathematical description of the road, provided by Sally, to construct an arc length integral and determine the distance that the road travels. You can then use the cross-sectional information provided by Naveen to compute the cross-sectional area with the help of Pappus' Centroid Theorem.

Estimating the Volume

To estimate the amount of concrete you need, you can use Pappus' Centroid Theorem, which is a way of calculating the volume formed when a shape of area A travels some distance L, forming a volume. (This is also useful in simplifying problems with volumes by rotation.)

(oops.)

Pappus' Centroid Theorem states that the volume V of a solid generated when a shape of area A travels over some distance L is equal to the area of the shape times the total distance traveled by the centroid of the shape.

Similarly, the second theorem of Pappus states that the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d_2 traveled by the lamina's geometric centroid x,

If the centroid of a cross-sectional shape travels a distance L to form a volume V, then the volume of concrete or asphalt is related to this distance and the cross-sectional area via:

$V=A*L$

where A is the cross-sectional area of the material (concrete or asphalt):

$A=\int _{a}^{b}f(x)-g(x)dx$

and is the area between two curves f(x) and g(x), and L is the length traveled by the centroid,

$L=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx$

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

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, Sally, your engineering co-worker who gave you the original map of the site runs up to you in a panic and apologizes, explaining that her map LEFT OUT CRITICAL INFORMATION.

Sally informs you that the road is actually sloped at a 4% grade the entire length of the road. That means, for every 100 feet the road travels, it rises 4 feet. This will affect the actual length of the road, and the amount of material you need.

You don't have your spreadsheets or calculations in front of you - all you have is a pencil, and the back of an envelope. If you want more material in time for the construction job, you'll have to order it before the dump trucks leave for the construction site in 15 minutes - the pressure is on. How much more concrete and asphalt do you need to order?

4) What is the approximate increase in the cost of concrete and asphalt for the construction job that resulted from Sally's mistake?