© 21St Century Math Projects

Home , Tunnel of Eupalinos

© 21st Century Math Projects Project Title: Tunnel of Eupalinos – Civil Engineering Standard Focus: Geometry & Spatial Sense, Time Range: 2-3 Days Measurement Supplies: Rulers, Protractors, Wax Paper, PlayDough, Toothpicks, Laser Pen or Flashlight. Maybe cardboard. Topics of Focus: - Law of Sines

- Measurement

Benchmarks: 11. (+) Understand and apply the Law of Sines and the Law of Cosines to find Similarity, Right Triangles, G-SRT unknown measurements in right and non-right triangles (e.g., surveying problems, and Trigonometry resultant forces).

Procedures:

A.) Student will complete “Civil Engineer”. In this assignment, students will solve a variety of bridge and tunnel problems using advanced trigonometry. The problems range in difficulty. They may require to sketch pictures or make recommendations.

B.) Students will complete “Tunnel of Eupalinos”. Students watch a video about the Tunnel of Eupalinos and answer a few questions. This assignment would be brief and would only last 15 minutes or so. The link to the primary video is: https://www.youtube.com/watch?v=JPP2pdEI-lc It is included as a QR Code later for students. Other resources you could opt to use are: http://www.mamikon.com/TunnelSamos.pdf http://www.sciencechannel.com/tv-shows/what-the-ancients-knew/videos/what-the-ancients-knew- eupalinos-tunnel-like-water-f.htm’

C.) Students will then complete “Mount Playdough” which will ask students to design and construct a tunnel from both sides of a lump of Playdough. Further instructions are provided later in the assignment. Wax paper is useful because students can sketch on it and it will keep the Playdough as neat as possible. Students will need to have a partner and the goal is that from opposites sides of the mountain of PlayDough that they can design, sketch and calculate a triangle to keep their construction efforts on task. When students have finished their tunnels there are two tests. One is whether or not a laser pen can shine all the way through it. Two is whether water can flow from one end to the other.

If you’d like to make this an event at the end of class, I suggest getting each set of students a piece of cardboard so they can carry their Playdough Mountain to a central space in the classroom so it can be tested in front of an audience 

© 21st Century Math Projects Building bridges and digging tunnels fall under the purview of a civil engineer. A civil engineer specializes in design and construction of things like roads, canals and buildings. Unlike a typical road, if a bridge must be built across a span or a tunnel must be dug through a hill or a mountain, direct measurements cannot be taken. In these cases, these engineering challenges will require advanced trigonometry and careful analysis.

In these problems, use the illustrations or sketch your own to calculate lengths and make decisions.

1.) A new highway is being built to better connect a growing suburb with downtown. It is required that this bridge crosses a river. To begin preliminary plans, a design firm needs to be able to measure the distance across the river from Point A to Point C. The firm began at Point A, and traveled 500 meters to Point B. The angle formed at angle A was 53o and the angle formed at angle B was 100o. Approximately, what is the distance from Point A to Point C?

2.) To estimate the maximum length of an oblong pond from Point A to Point C, a surveyor starts at Point A and walks 2,400 feet southwest to Point B. The angle formed at angle A was 108o degrees and the angle formed at angle B was 52o degrees. Approximate the maximum distance across the pond.

3.) In order to alleviate congestion for a highway system, there is a plan to dig a tunnel through a mountain. The city planners need to be able to measure the distance through the mountain from Point A to Point C. The firm began at Point A, and traveled 3.5 miles to Point B. The angle formed at angle A was 66o and the angle formed at angle B was 77o. Approximately, what is the distance from Point A to Point C?

4.) In preparations for an upcoming sewer project that will tunnel through a large elliptical hill, engineers are attempting to measure the shortest length from Point A to Point C. A surveyor starts at Point A and walks 12,500 feet northeast to Point B. The angle formed at angle A was 65o degrees and the angle formed at angle B was 69o degrees. Approximate the minimum distance through the hill.

© 21st Century Math Projects 5.) A tourism company plans to build a bridge from land to a nearby island to increase potential guests. From the mainland, the only two places that the bridge can be started (at Point A and Point B) are 5000 feet apart. To help decide between Bridge A (which connects A to C) or Bridge B (which connects B to C), the company will need precise lengths of the two possible bridges. The angle formed at angle A was 62.5o and the angle formed at angle B was 35.9o. Determine the approximate lengths of each bridge.

5a.) Due to specific requirements that each bridge would require, it is estimated that Bridge A would cost $560 per foot and Bridge B would cost $490 per foot. How much would it cost to build each bridge? What would be the savings if the less expensive bridge was chosen?

6.) A civil engineer has been tasked with putting forward a plan to build a bridge for the Kentucky Lake Bridges Project. The engineer has estimated costs for each project and has performed measurements in an effort to estimate the lengths of each project. Help the engineer finalize her report and make a recommendation.

© 21st Century Math Projects 7.) To better connect its citizens, a country has moved forward with plans for a high-speed rail system. In order to complete the project, a tunnel will need to be dug through a mountain. Based on current rail plans there are two possible places that the tunnel can be started (at Point A and Point B) that are 15,000 meters apart. To help decide between Tunnel A (which connects A to C) and Tunnel B (which connects B to C), the design team will need precise lengths of the two possible tunnels. The angle formed at angle A was 80.2o and the angle formed at angle B was 40.1o. Determine the approximate lengths of each tunnel.

7a.) Due to the type of digging that would be required and the predicted mineral composition of the different parts of the mountain, it is estimated that Tunnel A would cost $4,580 per meter and Tunnel B would cost $3,875 per meter. How much would it cost to construct each tunnel? What would be the savings if the less expensive tunnel was chosen?

8.) A civil engineer has been tasked by the city of Sacramento to design a tunnel for the Delta Habitat Conservation Program to ensure reliable water deliveries through the state. The engineer has estimated costs for each project and has performed measurements in an effort to estimate the lengths of each tunnel option. Help the engineer finalize his report and make a recommendation.

In ancient warfare, an aggressive and effective tactic is to cut off your enemy from their water supply. In 6 BC, on the island of Samos in Greece conflict was often. In an effort to stay connected to their water supply, the Greeks directed an engineer Eupalinos to tunnel through Mount Kastro to build an An aqueduct is a water path that aqueduct. The Greeks believed that an underground was typically constructed to carry aqueduct would not be able to be found by their enemies water from a source to a city. and would be protect the citizens.

Watch the History Channel segment on the Tunnel and answer the questions.

1. Who was the leader of Samos that directed Eupalinos to build the tunnel?

2. Given the size of this project, how did they decide to dig the tunnel?

3. What challenges were presented in the project?

Sketch the strategy that Eupalinos used onto the map of Mount Kastro below:

Tunneling is difficult enough, but like with the Tunnel of Eupalinos digging both ends of the tunnel at the same time is quite a challenge. In this project you will channel your inner Eupalinos with a partner. Given one lump of Playdough, wax paper, a ruler, a protractor and two toothpicks your task is to design and construct a tunnel.

You and your partner must be on opposite sides of the Playdough lump and cannot peek to the other side after construction begins. You must choose a degree of difficulty: Ground, Elevated or Aqueduct You must construct a triangle on the wax paper with side lengths and angle measures calculated. You may only use the toothpick to remove Playdough. If your structure falls, you must start all over again. Light Test: A laser pen must be able to pass through the tunnel. Water Test: If water can be dropped into the entrance of the tunnel and passes to the exit, bonus points will be earned.

Team grades for the project will be broken into the following components

Degree of Difficulty

Aqueduct: an elevated sloping tunnel that intends to carry water from a source to a city. ___ /110 pts 100 pts Elevated: a tunnel that is in the middle of the Playdough and the table is not visible in the tunnel. ___ /100 pts Ground: a tunnel that is at the base of the Playdough and the table is visible throughout. ___ /90 pts Research & Design 200 pts triangle was sketched, measured and calculated appropriately ____/200 pts Results (the tunnel…)

passed the Light and Water test. 120 pts passed the Light or Water test 100 pts 100 pts did not pass the Light or Water test, but both ends of the tunnel meet. 60 pts was not completed, but a valiant attempt was made 50 pts was not completed 0 pts

Total _____/400

Thank you for being my Math Friend!

If you liked this 21st Century Math Project You might like others. (Click the logo)

Math it Up. Boomdiggy.

In these problems, use the illustrations or sketch your own to calculate lengths and make decisions.

The distance would be 1084.6 meters.

The bridge would be 5529.8 feet.

The tunnel would be 5.67 miles.

The tunnel would be 16,222.87 feet long.

© 21st Century Math Projects 5.) A tourism company plans to build a bridge from land to a nearby island to increase potential guests. From the mainland, the only two places that the bridge can be started (at Point A and Point B) are 5000 feet apart. To help decide between Bridge A or Bridge b, the company will need precise lengths of the two possible bridges. The angle formed at angle A was 62.5o and the angle formed at angle B was 35.9o. Determine the approximate lengths of each bridge.

Bridge a would be 2963.6 ft Bridge b would be 4483.1 ft.

5a.) Due to specific requirements that each bridge would require, it is estimated that Bridge a would cost $560 per foot and Bridge b would cost $490 per foot. How much would it cost to build each bridge? What would be the savings if the less expensive bridge was chosen?

4483.1 * 490 = 2,196,719 for Bridge B and 2963.6*560 = 1,659,616 for Bridge A. Bridge B would cost 537,103 less.

© 21st Century Math Projects 7.) To better connect its citizens, a country has moved forward with plans for a high-speed rail system. In order to complete the project, a tunnel will need to be dug through a mountain. Based on current rail plans there are two possible places that the tunnel can be started (at Point A and Point B) that are 15,000 meters apart. To help decide between Tunnel a and Tunnel b, the design team will need precise lengths of the two possible tunnels. The angle formed at angle A was 80.2o and the angle formed at angle B was 40.1o. Determine the approximate lengths of each tunnel.

Tunnel a would be 17,119.8 m. Tunnel b is 11,190.5 m

17,119.8*3875 = $66,339,225 for Tunnel B and 51,252,490 for Tunnel A. Tunnel A would save $15,086,735.

Watch the History Channel segment on the Tunnel and answer the questions.

1. Who was the leader of Samos that directed Eupalinos to build the tunnel? Polycrates

2. Given the size of this project, how did they decide to dig the tunnel? From both ends.

3. What challenges were presented in the project? Meet in the same horizontal and vertical height.

Sketch the strategy that Eupalinos used onto the map of Mount Kastro below: