Worksheets/Civil Engineering Road Planning - Revision history

Admin: /* Estimating the Volume */

2019-12-09T20:39:53Z

Estimating the Volume

← Older revision		Revision as of 20:39, 9 December 2019
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	(oops.)		(oops.)

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

	~~states that 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:

	<math>		<math>

Admin: /* Your Job */

2019-12-09T20:38:37Z

Your Job

← Older revision		Revision as of 20:38, 9 December 2019
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	==Your Job==		==Your Job==

	Your job in the construction process is to estimate the total volume of concrete ~~and~~ 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.		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==		==Estimating the Volume==

Admin: /* Calculation: Area and Volume */

2019-12-09T20:38:19Z

Calculation: Area and Volume

← Older revision		Revision as of 20:38, 9 December 2019
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	<math>		<math>
	A = int_{a}^{b} ( f(x) - g(x) ) dx		A = \int_{a}^{b} ( f(x) - g(x) ) dx
	</math>		</math>

Admin: /* Your Task */

2016-05-22T01:52:08Z

Your Task

← Older revision		Revision as of 01:52, 22 May 2016
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	==~~Your Task~~==		==Calculation: Area and Volume==

	~~Your job in the construction process is to estimate the total volume~~ of ~~concrete and 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~~. Using Pappus' Centroid Theorem to estimate the total volume~~ of the ~~concrete and asphalt material on the highway.~~		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:

			<math>
			A = int_{a}^{b} ( f(x) - g(x) ) dx
			</math>

			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:

			<math>
			V = A L
			</math>

			==Your Job==

			Your job in the construction process is to estimate the total volume of concrete and 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==		==Estimating the Volume==

Admin: /* Worksheet Contents */

2016-05-22T01:02:07Z

Worksheet Contents

← Older revision		Revision as of 01:02, 22 May 2016
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	Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.		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==		==Calculation: Road Length==

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

	<math>		<math>
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	</math>		</math>

			This will require you to create a coordinate system and describe the curve and two straight sections as a function <math>f(x)</math>.
	~~==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~~.

	==Information from Naveen: Cross-Section of Road==		==Information from Naveen: Cross-Section of Road==

Admin: /* Information from Sally: Road Coordinate System */

2016-05-22T00:59:34Z

Information from Sally: Road Coordinate System

← Older revision		Revision as of 00:59, 22 May 2016
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	Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.		Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.

			==Calculation: Road Length==

			To calculate the length of the road, use the arc length formula:

			<math>
			L = \int_{a}^{b} \sqrt{ 1 + (f'(x))^2 } dx
			</math>


	==Information from Sally: Road Coordinate System==		==Information from Sally: Road Coordinate System==

Admin: /* Problem setup */

2016-05-22T00:58:32Z

Problem setup

Admin: /* The Problem */

2016-05-22T00:57:50Z

The Problem

← Older revision		Revision as of 00:57, 22 May 2016
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	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.		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.

	=~~The Problem~~=		==Information from Naveen: Cross-Section of Road==

	~~==Route Layout==~~		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.

	~~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.~~		(Cross-section of road bed on straight sections)

	~~...~~		(Cross-section of road bed on curved sections)

	(Explanation of drawing)		(Explanation of drawing)

	~~Information provided:~~
	* Town A coordinates
	* Town B coordinates
	* X miles from Town A, bearing X degrees NE
	* X miles from Town B, bearing X degrees NW
	* Curvature of circle

	~~How to specify the circle?~~

	==~~Road Design~~==		==Your Task==

	~~Naveen provides~~ you ~~with~~ the cross-sectional ~~design of~~ the ~~road. The~~ cross-sectional ~~design~~ of the ~~road incorporates elements like sloped surfaces to prevent water buildup,~~ and the ~~ability to expand or contract during heating and cooling.~~		Your job in the construction process is to estimate the total volume of concrete and 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. Using Pappus' Centroid Theorem to estimate the total volume of the concrete and asphalt material on the highway.

	...

	~~(Cross-section of road bed on straight sections)~~

	~~...~~

	~~(Cross-section of road bed on curved sections)~~

	~~...~~

	~~(Explanation of drawing)~~

	~~==Your Task==~~

	Your job in the construction process is to estimate the total volume of concrete and 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. Using Pappus' Centroid Theorem to estimate the total volume of the concrete and asphalt material on the highway.

	==Estimating the Volume==		==Estimating the Volume==

Admin: /* Road Coordinate System */

2016-05-22T00:55:29Z

Road Coordinate System

← Older revision		Revision as of 00:55, 22 May 2016
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	Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.		Naveen is giving us the design of the road's cross-section. We can use that to obtain an area.

	==Road Coordinate System==		==Information from Sally: Road Coordinate System==

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

Admin: /* Problem Procedure */

2016-05-22T00:55:15Z

Problem Procedure

← Older revision		Revision as of 00:55, 22 May 2016
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	To estimate how much concrete and asphalt to order, we need to know the volume.		To estimate how much concrete and asphalt to order, we need to know the volume.

	~~The~~ volume ~~is related to the~~ area ~~(cross-sectional area)~~ and length ~~of the road~~		To find a volume, we need an area and a length.

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

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

	~~Volume can be obtained from area and length~~

	==Road Coordinate System==		==Road Coordinate System==

← Older revision		Revision as of 00:58, 22 May 2016
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	* That will allow us to calculate the volume of materials required to build it		* That will allow us to calculate the volume of materials required to build it

	Coordinate System:		Information from Sally: Coordinate System
	* information provided by colleague to describe the road quantitatively		* information provided by colleague to describe the road quantitatively
	* circular part, two linear parts		* circular part, two linear parts
	* make up our own coordinate system		* make up our own coordinate system

	Road Length:		Calculation: Road Length
	* Arc length calculation, review arc length formula		* Arc length calculation, review arc length formula
	* Two lines, one circle - how to turn a circle into a function?		* Two lines, one circle - how to turn a circle into a function?

	Road Cross-Section:		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.		* 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		* Our new highway has 4 lanes, total width of 4 * 15 feet + 2 * 10 feet = 60 + 20 = 80 feet

	~~Road Volume~~:		Calculation: Areas and Volumes
	* Pappus' Theorem: center of gravity, V = A L		* Pappus' Theorem: center of gravity, V = A L