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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