This is a project. Below is an attachment with all the necessary information on what to do. There are instructions and at the end there is one example for each Activity. There are three activities.Great Circle Routes

Consider this strange truth about geography: Tokyo is about 400 miles further south than

Chicago, and Chicago is about 1300 miles further south than Anchorage, Alaska. So why does

the shortest airline route between Chicago and Tokyo take the plane north of Anchorage,

Alaska?

This is the first of three class activities in this semester of the course. A passing score on this

assignment is required for you to receive credit for this course. The purpose of this assignment

is to show you how trigonometry is used to determine the shortest routes when flying very long

(intercontinental) distances.

Directions:

Below you will find five different threads started by your teacher. These threads have subject

headings of “Introduction to Great Circle Routes”, “Latitudes and Longitudes of Cities”,

“Activity #1: The Longitude Challenge”, “Activity #2: The Latitude Challenge”, and “Activity

#3: The Opposite Hemisphere Challenge”.

1. Begin by reading the “Introduction to Great Circle Routes” thread, and the “How to

Generate Greek Characters” reply posted to it. If you have any questions about the

concept of a great circle route, or how to work with the equations, please post your

questions in this thread.

2. Follow the instructions for the “Activity #1: The Longitude Challenge”. Use the

coordinates listed in the “Latitudes and Longitudes of Cities” thread to do this and the

other activities. You must show your work using the Equation Editor. See the sample

problems posted by Mr. Hennon for a good example of what “showing your work” looks

like for this activity. Post your assignment for each activity as a reply to your teacher’s

post. Do not post your assignments as replies to other students’ posts.

3. Follow the instructions for the “Activity #2: The Latitude Challenge”. Again, you must

show your work using the Equation Editor. Post your assignment for each activity as a

reply to your teacher’s post. Do not post your assignments as replies to other students’

posts.

4. Finally, follow the instructions for the “Activity #3: The Opposite Hemisphere

Challenge.” As with the other two activities, you must show your work using the

Equation Editor. Post your assignment for each activity as a reply to your teacher’s post.

Do not post your assignments as replies to other students’ posts.

Grading:

This assignment has 100 possible points and is divided into three smaller activities:

Activity #1: The Longitude Challenge (30 points)

Activity #2: The Latitude Challenge (30 points)

Activity #3: The Opposite Hemisphere Challenge (40 points)

A passing score on this assignment is required for you to receive credit for this course.

You may have heard your Geometry course referred to at some point as “Euclidean” geometry.

This name comes from the ancient Greek mathematician Euclid, who in about 300 B.C.

compiled and organized most of the mathematics known to the Greeks, Romans, Egyptians, and

Babylonians into a comprehensive work called The Elements. Euclidean geometry is built up

from a few basic principles assumed to be true.

The problem is, however, that sometimes these principles are not true after all. One of these

times is when we are dealing with the surface of a sphere instead of a truly flat surface. For

example, you already know that the angles of a triangle must add up to 180 degrees, right? Not

necessarily so on the surface of a sphere! Consider a huge triangle on the surface of the Earth.

Put one vertex of this triangle at the North Pole, and put the other two on the equator.

Specifically, put one vertex at the point where the Prime Meridian crosses the Equator (it will be

in the Atlantic Ocean about 400 miles south of the African country of Ghana), and put the other

vertex where 90° West longitude crosses the Equator (it will be in the Galapagos Islands in the

Pacific Ocean, about 700 miles west of the South American country of Ecuador). What we have

here is a triangle that covers 1/8th of the surface of the Earth, and whose angles add up not to

180°, but to 270°. This is a triangle made up of 3 right angles! Geometries in which the usual

Euclidean rules no longer apply are called, appropriately enough, “Non-Euclidean” geometries.

Over large distances on the surface of a sphere, the shortest distance between two points is not a

straight line, but rather a curved line. These curved lines are called “Great Circles”. To see some

great circles for yourself, download a program like Google Earth and use the ruler tool to

measure the distances between far-flung global cities. Google Earth is especially good for this

because it automatically connects the two endpoints with a great circle. In particular, try to find

cities of similar latitude and look how the shortest path deviates to the north (in the northern

hemisphere) or the south (in the southern hemisphere). As mentioned above, note how the great

circle between Chicago and Tokyo goes way up north into Alaska. For some really dramatic

examples, note how a direct flight from Mumbai, India to Mexico City takes you over the Arctic

Ocean, while a flight from Buenos Aires, Argentina to Sydney, Australia almost takes you over

Antarctica.

Some examples of great circle routes can be seen in the “Sample Problems” thread.

To find the distance between two cities along a great circle route, we need to know the latitude

(theta) and the longitude

them

(phi) for each city, as well as the difference in longitude between

(pronounced “delta phi”) For example, if one city is at 90° West longitude and the

other is at 65° West longitude, then

calculating

equals 90° – 65° = 25°. Be careful when

when your cities are on opposite ends of the Prime Meridian. For example, if

one city is at 80° West longitude and the other is at 20° East longitude, then

is not equal to

80° – 20° = 60°. Rather, it is equal to 80° – (-20°) = 100°. It might help for you to consider one

side of the Prime Meridian as being “positive” longitude while the other side as being “negative”

longitude.

The values for

will range from 0 to 180°. If you have initially calculated a

greater than

180° (for example, you found that the distance between a city at 120°W and 100°E was 220°),

then subtract that number from 360 and use the difference as your

360° – 220° = 140°, and so the

value. (For the example,

for the cities at 120°W and 100°E will be 140°)

With regards to your latitude values of , use positive numbers for latitudes north of the equator,

and negative numbers for latitudes south of the equator.

Finally, remember to convert all angle values to radians instead of degrees.

To actually calculate the distance between two cities, here is the formula:

Suppose City #1 has latitude

longitude

and longitude

. Let the radius of the Earth be

miles. The distance

, and that City #2 has latitude

and

, which for the purposes of this project is 3959

along the great circle route between City #1 and City #2 is defined as:

Other threads will give latitude and longitude coordinates for a variety of cities, and show you

how their distances are calculated.

–teacher

This thread contains latitudes and longitudes of over 40 major world cities. Remember to convert

the latitude and the longitude from degrees to radians when calculating the distance along the

great circle route.

City (Latitude, Longitude)

Amsterdam, The Netherlands (52.37°N, 4.90°E)

Atlanta, USA (33.76°N, 84.39°W)

Bangkok, Thailand (13.75°N, 100.47°E)

Barcelona, Spain (41.38°N, 2.18°E)

Beijing, China (39.92°N, 116.38°E)

Boston, USA (42.36°N, 71.06°W)

Brussels, Belgium (50.85°N, 4.35°E)

Buenos Aires, Argentina (34.60°S, 58.38°W)

Chicago, USA (41.84°N, 87.68°W)

Dubai, UAE (24.95°N, 55.33°E)

Dublin, Ireland (53.35°N, 6.26°W)

Frankfurt, Germany (50.12°N, 8.68°E)

Hong Kong, China (22.30°N, 114.20°E)

Istanbul, Turkey (41.01°N, 28.96°E)

Jakarta, Indonesia (6.20°S, 106.82°E)

Johannesburg, South Africa (26.20°S, 28.05°E)

Kuala Lumpur, Malaysia (3.13°N, 101.68°E)

London, UK (51.51°N, 0.13°W)

Los Angeles, USA (34.05°N, 118.25°W)

Madrid, Spain (40.40°N, 3.68°W)

Melbourne, Australia (37.81°S, 144.96°E)

Mexico City, Mexico (19.43°N, 99.13°W)

Miami, USA (25.78°N, 80.21°W)

Milan, Italy (45.47°N, 9.18°E)

Moscow, Russia (55.75°N, 37.62°E)

Mumbai, India (18.98°N, 72.83°E)

Munich, Germany (48.13°N, 11.57°E)

New Delhi, India (28.61°N, 77.21°E)

New York City, USA (40.71°N, 71.01°W)

Paris, France (48.86°N, 2.35°E)

Prague, Czech Republic (50.08°N, 14.42°E)

San Francisco, USA (37.78°N, 122.42°W)

Sao Paulo, Brazil (23.55°S, 46.63°W)

Seoul, South Korea (37.57°N, 126.97°E)

Shanghai, China (31.20°N, 121.50°E)

Singapore, Singapore (1.28°N, 103.83°E)

Stockholm, Sweden (59.33°N, 18.07°E)

Sydney, Australia (33.87°S, 151.21°E)

Taipei, Taiwan (25.03°N, 121.63°E)

Tokyo, Japan (35.68°N, 139.68°E)

Toronto, Canada (43.70°N, 79.40°W)

Vienna, Austria (48.20°N, 16.37°E)

Warsaw, Poland (52.23°N, 21.02°E)

Washington D.C., USA (38.90°N, 77.02°W)

Zurich, Switzerland (47.37°N, 8.55°E)

–teacher

Pairs No Longer Available

These pairs of cities are no longer available to use in this assignment:

Sample Problem Pairs:

Boston, USA and Madrid, Spain

Chicago, USA and Tokyo, Japan

Hong Kong, China and Sydney, Australia

(You are welcome, however, to pair Boston, Chicago, Hong Kong, Madrid, Sydney, or Tokyo up

with other cities, provided no one else has already done so.)

Student-Used Pairs:

(None yet)

–teacher

Sample Problems

Sample Problem #1

Calculate the distance between Chicago, USA and Tokyo, Japan along the great circle route.

Round off your radian measures to 4 decimal places and your great circle distance to the nearest

10 miles.

Solution:

Let City #1 be Chicago. Since Chicago’s coordinates are 41.84°N, 87.68°W, this means

that

radians, and that

radians .

Let City #2 be Tokyo. Since Tokyo’s coordinates are 35.68°N, 139.68°E, this means

that

radians.

radians, and that

Since the longitudinal difference between Chicago and Tokyo was calculated to be over 180°

(87.68° + 139.68° = 227.36°), we will subtract 227.36° from 360° to

get

radians.

Remembering that the radius of the Earth is 3959 miles, we can now calculate the distance along

the great circle route from Chicago to Tokyo as being:

Therefore, the great circle route from Chicago to Tokyo is approximately 6300 miles.

Sample problem #2

Calculate the distance between Boston, USA and Madrid, Spain along the great circle route.

Round off your radian measures to 4 decimal places and your great circle distance to the nearest

10 miles.

Solution:

Let City #1 be Boston. Since Boston’s coordinates are 42.36°N, 71.06°W, this means

that

radians, and that

radians .

Let City #2 be Madrid. Since Madrid’s coordinates are 40.40°N, 3.68°W, this means

that

radians, and that

radians.

Since the longitudinal difference between Boston and Madrid was calculated to be less than

180°, we will simply subtract 3.68° from 71.06° to get

radians.

Remembering that the radius of the Earth is 3959 miles, we can now calculate the distance along

the great circle route from Boston to Madrid as being:

Therefore, the great circle route from Boston to Madrid is approximately 3400 miles.

Note that while this great circle route goes noticeably northward, it doesn’t go anywhere near as

far north as the great circle route from Chicago to Tokyo. This is because the longitudinal

difference between Boston and Madrid is so much less than the difference between Chicago and

Tokyo.

Sample problem #3

Calculate the distance between Hong Kong, China and Sydney, Australia along the great circle

route. Round off your radian measures to 4 decimal places and your great circle distance to the

nearest 10 miles.

Solution:

Let City #1 be Hong Kong. Since Hong Kong’s coordinates are 22.30°N, 114.20°E, this means

that

radians, and that

radians .

Let City #2 be Sydney. Since Sydney’s coordinates are 33.87°S, 151.21°E, this means

that

radians.

radians, and that

Since the longitudinal difference between Hong Kong and Sydney was calculated to be less than

180°, we will simply subtract 114.20° from 151.21° to get

radians.

Remembering that the radius of the Earth is 3959 miles, we can now calculate the distance along

the great circle route from Hong Kong to Sydney as being:

Therefore, the great circle route from Hong Kong to Sydney is approximately 4580 miles.

Any time a great circle route covers a large difference in latitude, the tendency of the route to

creep north or south is minimized. That is why this route looks more “normal” than the other

two. It is not really surprising that a flight from Hong Kong to Sydney would pass over the

Philippines and Indonesia.

You’ll also notice that the distances you calculate differ slightly from what you might find on

Google Earth. This is due to two reasons. First, the fact that we are rounding off our radian

measures adds a slight bit of error into our answers. Second, and more importantly, the Earth is

not, in fact, a perfect sphere. The Earth’s rotation causes the planet to bulge slightly at the

equator. Therefore, the distance from the center of the Earth outward to the equator is several

miles greater than the distance from the center of the Earth outward to the North (or South) Pole.

–teacher

How to Generate Greek Characters

This post shows you how to generate the Greek characters used in the assignment using the

Equation Editor:

Type this in the editor … to generate this.

theta_1

theta_2

phi_1

phi_2

Delta phi

(Note: Make sure you capitalize the D in “Delta”)

Also:

cos^-1(0.1742)

Example of activity 1

Great Circle Distance of Amsterdam, The Netherlands (52.37°N,

4.90°E); Bangkok, Thailand (13.75°N, 100.47°E)

Let City 1 be Amsterdam. Since Amsterdam’s coordinates are 52.37°N,

4.90°E, this means that radians, and that radians .

Let City 2 be Bangkok. Since Bangkok’s coordinates are 13.75°N,

100.47°E, this means that radians, and that radians.

radians.

Remembering that the radius of the Earth is 3959 miles, we can now

calculate the distance along the great circle route from Amsterdam to

Bangkok as being:

The great circle route from Amsterdam to Bangkok is 5703 miles.

Example of Activity 2

Great Circle Distance of Milan, Italy (45.47°N, 9.18°E); Singapore,

Singapore (1.28°N, 103.83°E)

Let City 1 be Milan. Since Milan’s coordinates are 45.47°N, 9.18°E, this

means that radians, and that radians .

Let City 2 be Singapore. Since Singapore’s coordinates are 1.28°N,

103.83°E, this means that radians, and that radians.

radians.

Remembering that the radius of the Earth is 3959 miles, we can now

calculate the distance along the great circle route from Milan to Singapore

as being:

The great circle route from Milan to Singapore is 6381 miles.

Example of Activity 3

Great Circle Distance of Atlanta, USA (33.76°N, 84.39°W); Sydney, Australia (33.87°S, 151.21°E)

Let City 1 be Atlanta. Since Atlanta’s coordinates are 33.76°N, 84.39°W, this means

that

radians, and that

Let City 2 be Sydney. Since Sydney’s coordinates are 33.87°S, 151.21°E, this means

that

that

radians, and

radians.

radians.

radians .

Remembering that the radius of the Earth is 3959 miles, we can now calculate the distance along the

great circle route from Atlanta to Sydney as being:

The great circle route from Atlanta to Sydney is 9289 miles.

(Given above are one example of each activity.

The instructions are at the top of the page.

Should contain different coordinates etc.

Change up the problems but try to keep it neat

and similar to the three examples just given.

As in the similarity should be how it is

structured. No need to include pictures. Just

math and explanation in a separate word

document.)

Purchase answer to see full

attachment

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