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Ahmed 1
Thanvir Ahmed
Mrs. Tallman
Pre-Calculus 11A
10 November 2014
Around the World
I have recently decided to go sight-seeing around the world. I have decided to visit some
of the strangest natural phenomena in the world. This journey will take me to the dry plateaus of
the Grand Canyon, to the snowy tops of the Himalayas to visit Mt. Everest, and finally to the
scorching fires of the Door to Hell. On this journey I will travel from Michigan to all of these
places in the order listed above.
I want to travel to these places in the shortest distance possible. In order to calculate this
distance I will need to use the Spherical Law of Cosines. To use the Spherical Law of Cosines I
have to calculate different angel measurements from the latitudes and longitudes. Below there is
a table portraying the different latitudes and longitudes for my journey.
Table 1 Latitude and Longitude
Latitude°,’,’’
Longitude°,’,’’
HomeWarren, Michigan 42° 28' 39" N 83° 1' 40" W
Destination 1The Grand Canyon 36° 14' 50" N 112° 20' 2" W
Destination 2Mt. Everest 27° 59' 10" N 86° 55' 21" E
Destination 3Door to Hell 40° 15' 9" N 58° 26' 22" E
The table above shows the latitude and longitude for all of the destinations and home.
These values are used to find angle measures to calculate the distance within the Spherical Law
of Cosines.
Ahmed 2
To begin my trip I will go from Warren, Michigan (Home) to the Grand Canyon
(Destination 1). The Grand Canyon is one of the most historic places on Earth. The layer of
rocks on the canyon walls helped determine the age of the Earth. The different layers tell stories
about the origins of the Earth and are embedded with fossil remains. The size of the Grand
Canyon is also overwhelming. The coordinates for Warren, Michigan are 42° 28' 39" N and 83°
1' 40" W. The coordinates for the Grand Canyon are 36° 14' 50" N and 112° 20' 2" W.
Figure 1. The Grand Canyon
The figure above shows the Grand Canyon and its overwhelming size. It has over 277
rivers, which are up to 18 miles wide and 1 mile deep.
Figure 2. Warren in Reference to the North Pole
Ø
42° 28' 39"
Warren, MI
North Pole
Ø=90°- 42° 28' 39"
Ø= 47.5225°
Ahmed 3
The figure above shows where Warren, Michigan is in reference to the North Pole. The
latitude of Warren is 42° 28' 39" N, which means that it is 42° 28' 39" above the equator. The
North Pole is perpendicular to the equator, which means that it forms a 90° with the equator. The
latitude of Warren is subtracted from 90°, in order to calculate the angle between the North Pole
and Warren Michigan, which is 47.5225°. This angle is equivalent to b in the equation shown in
Figure 6.
Figure 3. The Grand Canyon in Reference to the North Pole
The figure above shows where the Grand Canyon is in reference to the North Pole. The
latitude of the Grand Canyon is 36° 14' 50" N, which means that it is 36° 14' 50" above the
equator. The latitude subtracted from 90°, in order to calculate the angle between the North Pole
and the Grand Canyon, which is 53.7528°. This angle is equivalent to c in the equation shown in
Figure 6.
Ø36° 14' 50" Ø=90°- 36° 14' 50"
Ø=53.7528°
North Pole
Grand Canyon
Ahmed 4
Figure 4. Angle between Grand Canyon and Warren, Michigan
The figure above shows both Warren, Michigan and the Grand Canyon in reference to the
North Pole. The angle between Warren, Michigan and the Grand Canyon is acquired by
calculating the difference between the two. The angle that is formed with Warren, the North
Pole, and the Grand Canyon is 29.3061°. This is also A in the equation shown in Figure 6.
Figure 5. Warren, Michigan to the Grand Canyon
The figure above shows Warren, Michigan, the Grand Canyon, and the North Pole on a
sphere. The sphere is the earth. The distance between Warren, Michigan and the Grand Canyon
can be calculated using the Spherical Law of Cosines as shown in Figure 6.
North Pole
83° 1' 40"Warren, MI
Grand Canyon
112° 20' 2"
Ø
Ø=112° 20' 2"-83° 1' 40"
Ø= 29.3061°
Warren, MIGrand Canyon
North Pole
Distance
Ahmed 5
cos (a )=cos (b )∗cos (c )+sin (b )∗sin (c )∗cos (A )
cos (a )=cos (47.5225 ° )∗cos (53.7528 ° )+sin (47.5225 ° )∗sin (53.7528 ° )∗cos (29.3061° )
cos (a )=0.9180
(a )=cos−1(0.9180)
(a )=23.3691 °
distance=23.3691°360 °
∗2π (3960)
distance=1,615.1562miles
Figure 6. Distance between Warren, Michigan and the Grand Canyon
The figure above shows the distance between Warren, Michigan and the Grand Canyon.
The Spherical Law of Cosines is used to calculate this. The angle values that were used were
calculated in Figures 2, 3, and 4. The value of a was calculated to equal 23.3691°, but in order to
find the distance it must be divided by 360° and multiplied by 2π(3960). This is because the
Earth is a sphere, measuring 360°. The circumference of the Earth is 2π(3960) miles. The
distance between Warren, Michigan and the Grand Canyon was calculated to equal 1,615.1562
miles. This means that along the Great Circle going through both Warren and the Grand Canyon
the distance is 1,615.1562 miles.
Next, my trip will take me from the Grand Canyon to Mt. Everest (Destination 2). Mt.
Everest is the living embodiment of Plate Tectonics. Mt. Everest was created as a result of the
Indian subcontinent plate colliding in with Eurasian continent plate. This charge was at rates of
up to 15 cm/year. This is relatively slow, but over the course of millions of years Mt. Everest
was able to rise. Today it is the highest mountain on Earth, with a height of 29,028 ft. (5.5
miles). The coordinates for Mt. Everest are 27° 59' 10" N and 86° 55' 21" E. The angle the
Grand Canyon makes with the North Pole was calculated to be 53.7528° in Figure 3. This angle
is equivalent to b in the equation shown in Figure 11.
Ahmed 6
Figure 7. Mt. Everest
The figure above shows Mt. Everest. It is the highest mountain in the world. It is found
within the Himalayan Mountain Range.
Figure 8. Mt. Everest in Reference to the North Pole
The figure above shows Mt. Everest in reference to the North Pole. The latitude of Mt.
Everest is 27° 59' 10"N, which means that it is 27° 59' 10" above the equator. The latitude of Mt.
Everest is subtracted from 90°, in order to calculate the angle between the North Pole and Mt.
Everest which is 62.0138°. This angle is equivalent to c in the equation shown in Figure 11.
Ø
Mt. Everest
North Pole
Ø=90°- 27° 59' 10"
Ø= 62.0138°27° 59' 10"
Ahmed 7
Figure 9. Angle between Grand Canyon and Mt. Everest
The figure above shows both the Grand Canyon and Mt. Everest in reference to the North
Pole. The angle between the Grand Canyon and Mt. Everest is acquired by calculating the sum of
the two longitudes. The angle that is formed with the Grand Canyon, the North Pole, and
Mt. Everest is 199.2564°. This is also A in the equation shown in Figure 11.
Figure 10. The Grand Canyon to Mt. Everest
The figure above shows the Grand Canyon, Mt. Everest, and the North Pole on a sphere
(Earth). The distance between the Grand Canyon and Mt. Everest can be calculated using the
Spherical Law of Cosines as shown in Figure 11.
North Pole86° 55' 21"
Grand Canyon
112° 20' 2"
Ø
Ø=112° 20' 2"+86° 55' 21"
Ø= 199.2564°
Mt. Everest
Grand Canyon
Mt. Everest
North Pole
Distance
Ahmed 8
cos ( a )=cos (b )∗cos (c )+sin (b )∗sin (c )∗cos (A )
cos (a )=cos (53.7528° )∗cos (62.0138 ° )+sin (53.7528° )∗sin (62.0138° )∗cos (199.2564 ° )
cos (a )=−0.3949
(a )=cos−1(−0.3949)
(a )=113.2574°
distance=113.2574°360 °
∗2 π (3960)
distance=7,827.7895miles
Figure 11. Distance between the Grand Canyon and Mt. Everest
The figure above shows the distance between the Grand Canyon. The Spherical Law of
Cosines is used to calculate this. The angle values that were used were calculated in Figures 3, 8,
and 9. The value of a was calculated to equal 113.2574°, but in order to find the distance it must
be divided by 360° and multiplied by 2π(3960). The distance between the Grand Canyon and Mt.
Everest was calculated to equal 7,827.7895 miles. This means that along the Great Circle going
through both the Grand Canyon and Mt. Everest the distance is 7,827.7895 miles.
My adventure will now take me from Mt. Everest to the Door to Hell (Destination 3). The
Door to Hell is a 230 ft. wide crater in the middle of the desert near the village of Deweze,
Turkmenistan. It was a natural gas reserve. The soviets set fire to it hoping that the gas would
burn out in 1971, but it still burns to this day. The coordinates for the Door to Hell are 40° 15' 9"
N and 58 26' 22" E. The angle Mt. Everest makes with the North Pole was calculated to be
62.0138° in Figure 8. This angle is equivalent to b in the equation shown in Figure 16.
Ahmed 9
Figure 12. Door to Hell
The figure above shows the Door to Hell. It has been burning for more than 40 years. The
gigantic bowl of flames is noted as one of the scariest places on Earth by the Huffington Post.
Figure 13. Door to Hell in Reference to the North Pole
The figure above shows the Door to Hell in reference to the North Pole. The latitude of the
Door to Hell is 40° 15' 9" N, which means that it is 40° 15' 9" above the equator. The latitude of
the Door to Hell is subtracted from 90°, in order to calculate the angle between the North Pole
and the Door to Hell which is 49.7475°. This angle is equivalent to c in the equation shown in
Figure 16.
North Pole
40°15' 9" Ø
Ø=90°- 40°15' 9"
Ø=49.7475°
Door to Hell
Ahmed 10
Figure 14. Angle between Mt. Everest and the Door to Hell
The figure above shows both Mt. Everest and the Door to Hell in reference to the North
Pole. The angle between Mt. Everest and the Door to Hell is acquired by calculating the
difference between the two longitudes. The angle that is formed with Mt. Everest, the North
Pole, and the Door to Hell is 28.4831°. This is also A in the equation shown in Figure 16.
Figure 15. Mt. Everest to the Door to Hell
The figure above shows Mt. Everest, the Door to Hell, and the North Pole on a sphere
(Earth). The distance between Mt. Everest and the Door to Hell can be calculated using the
Spherical Law of Cosines as shown in Figure 16.
North Pole86° 55' 21"
Door to Hell
58° 26' 22"
Ø
Ø=86° 55' 21"- 58° 26' 22"
Ø= 28.4831°
Mt. Everest
Door to Hell
Mt. Everest
North Pole
Distance
Ahmed 11
cos (a )=cos (b )∗cos (c )+sin (b )∗sin (c )∗cos (A )
cos (a )=cos (62.0138° )∗cos (49.7475 ° )+sin (62.0138 ° )∗sin ( 49.7475° )∗cos (28.4831 ° )
cos (a )=0.8956
(a )=cos−1(0.8956)
(a )=26.4152°
distance=26.4152°360 °
∗2π (3960)
distance=1,825.6887miles
Figure 16. Distance between Mt. Everest and the Door to Hell
The figure above shows the distance between Mt. Everest and the Door to Hell. The
Spherical Law of Cosines is used to calculate this. The angle values that were used were
calculated in Figures 8, 13, and 14. The value of a was calculated to equal 26.4152°, but in order
to find the distance it must be divided by 360° and multiplied by 2π(3960). The distance between
Mt. Everest and the Door to Hell was calculated to equal 1,825.6887 miles. This means that
along the Great Circle going through both the Mt. Everest and the Door to Hell the distance is
1,825.6887 miles.
I have now been to all three of my chosen destinations. I must now return to Warren,
Michigan from the Door to Hell. The distance between the Door to Hell and Warren, Michigan is
calculated in Figure 18. The angle that the Door to Hell makes with the North Pole is calculated,
in figure 13, to equal 49.7475°. This is also b in the equation shown in figure 18. The angle that
Warren, Michigan makes with the North Pole is calculated, in figure 2, to equal 47.5225°. This is
also c in the equation shown in figure 19.
Ahmed 12
Figure 17. Angle between the Door to Hell and Warren, Michigan
The figure above shows both the Door to Hell and Warren in reference to the North Pole.
The angle between the Door to Hell and Warren is acquired by calculating the sum of the two
longitudes. The angle that is formed with the Door to Hell, the North Pole, and the Warren is
141.4672°. This is also A in the equation shown in Figure 19.
Figure 18. The Door to Hell to Warren, Michigan
The figure above shows the Door to Hell, Warren and the North Pole on a sphere (Earth).
The distance between the Door to Hell and Warren, Michigan can be calculated using the
Spherical Law of Cosines as shown in Figure 19.
Door to Hell
Distance
Warren, Michigan
North Pole
83° 1' 40"
Door to Hell
58° 26' 22"
Ø
Ø=58° 26' 22"+83° 1' 40"
Ø= 141.4672°
North Pole
Warren, Michigan
Ahmed 13
cos (a )=cos (b )∗cos (c )+sin (b )∗sin (c )∗cos (A )
cos (a )=cos (49.7475 ° )∗cos (47.5225 ° )+sin (49.7475 ° )∗sin (47.5225 ° )∗cos (141.4672° )
cos (a )=−0.0040
(a )=cos−1(−0.0040)
(a )=90.2278 °
distance=90.2278 °360 °
∗2π (3960)
distance=6,236.0979miles
Figure 19. Distance between the Door to Hell and Warren, Michigan
The figure above shows the distance between the Door to Hell and Warren, Michigan.
The Spherical Law of Cosines is used to calculate this. The angle values that were used were
calculated in Figures 2, 13, and 17. The value of a was calculated to equal 90.2278°, but in order
to find the distance it must be divided by 360° and multiplied by 2π(3960). The distance between
the Door to Hell and Warren was calculated to equal 6,236.0979 miles. This means that along the
Great Circle going through both Door to Hell and Warren, Michigan the distance is 6,236.0979
miles.
This brings an end to my journey. I have visited the Grand Canyon, Mt. Everest, and the
Door to Hell. My journey began and ended in Warren, Michigan. In order to travel to these
locations I wanted to travel the smallest distance possible. In order to do this I had to travel along
the Great Circles on the Earth. The distances I traveled were calculated using the Spherical Law
of Cosines. The distance between Warren, Michigan and the Grand Canyon was calculated to
equal 1,615.1562 miles. The distance between the Grand Canyon and Mt. Everest was calculated
to equal 7,827.7895 miles. The distance between Mt. Everest and the Door to Hell was
calculated to equal 1,825.6887 miles. The distance between the Door to Hell and Warren was
calculated to equal 6,236.0979 miles. The total distance traveled was 17,504.7323 miles.
Ahmed 14
Works Cited
Bilham, Rodger. "Birth of the Himalaya." PBS. PBS, Nov. 2010. Web. 03 Nov. 2014.
<http://www.pbs.org/wgbh/nova/everest/earth/birth.html>.
David, Zwiefelhofer B. "Grand Canyon National Park." GrandCanyonNationalPark. FSBO,
n.d. Web. 02 Nov. 2014. <http://www.findlatitudeandlongitude.com/?
loc=grand+canyon+national+park>.
"Grand Canyon South Rim Bus Tour with IMAX Tickets." GrayLine. Gray Line Corp., n.d.
Web. 02 Nov. 2014. <http://www.grayline.com/tours/las-vegas/grand-canyon-south-
rim-bus-tour-with-imax-tickets-5862_2/>.
"The Latitude and Longitude of Warren, Michigan Is:." Warren,MichiganLatitude/longitude.
Travel Math, n.d. Web. 01 Nov. 2014. <http://www.travelmath.com/cities/Warren,
+MI>.
"Mount Everest Lat & Long." MountEverestMapLatLongCoordinates. © 2012-2014
Www.LatLong.net, n.d. Web. 03 Nov. 2014. <http://www.latlong.net/place/mount-
everest-14.html>.
United States. National Park Service. "Grand Canyon National Park (U.S. National Park
Service)." NationalParksService. U.S. Department of the Interior, 01 Nov. 2014.
Web. 01 Nov. 2014. <http://www.nps.gov/grca/index.htm>.
Visser, Nick. "The Door To Hell Is Possibly The Creepiest Place On Planet Earth (PHOTOS)."
TheHuffingtonPost. TheHuffingtonPost.com, 20 Nov. 2013. Web. 04 Nov. 2014.
<http://www.huffingtonpost.com/2013/11/20/door-of-hell_n_4311694.html>.
