Regional Team Problem
March 21, 2001
There are many ways to depict the surface of the earth on a flat map. In the 1500's Gerard Mercator drew one of the first flat maps. He stretched the polar regions both vertically and horizontally. This became known as the Mercator Projection.
Since then there have been many other types of projections. You have been provided with a "cylindrical projection."
You have three tasks that all apply to Greenland.
Your entire set of answers must fit on the front of the answer page and include any and all explanations and mathematical justifications.
The Scoring:
You will be scored on a four-point scale for each bulleted area:
Understands and Applies Mathematical Concepts and Procedures
• Geometric Sense: Shows understanding of scale drawings.
• Number Sense: Shows correct applications of procedures to deal with the distortion.
• Measurement: Uses estimation to obtain reasonable approximations.
Uses Mathematical Reasoning
• Supports conclusions with applicable mathematics and clear explanations.
Communicates the Results
• Represents the mathematical processes and ideas in an effective format.
Problem Solving
• Shows understanding of the problem.
• Correctly completes all of the required tasks.
Total Points
Area of Greenland Solutions
First Method:
Earth’s Radius=R=6670 km.
Earth’s Circumference=C=2_(6670 km.)=41,888 km.
Number of km/ten degrees at equator=41,888/30=1164 km.
2.7 cm on the Enlarged Greenland map is ten degrees so
431 km./cm
North to south is 6.5 cm
6.5 cm(431 km/cm)=2802 km in length, top to bottom.
The radii for the circles at latitudes of 60°, 70°, and 80° are R cos(60°), R cos(70°), and R cos(80°) or 3335 km, 2281 km, and 1158 km. The circumferences (using 2_r with _=3.14) are 20944 km,14325 km, and 7272 km.
However for our purposes the scale of 431 km/cm can be converted by multiplying by the cosines giving at 60° 215 km/cm, 70° 147km/cm, and at 80° 74.8 km/cm.
If rectangles A, B, and C are drawn as shown to approximate the area, their dimensions are:
A (11 cm x 2.4 cm) which converts:
(11 cm)(74.8 km/cm) = 823 km
(2.4 cm)(431 km/cm) = 1025 km
Area = 823 km x 1025 km = 843,575 sq km
B (7.4 cm x 2.5 cm) which converts:
(7.4 cm)(147 km/cm) = 1088 km
(2.5 km)(431 km/cm) = 1077 km
Area = 1088 km x 1077 km = 1,171,776 sq km
C (2.1 cm x 1.3 cm) which converts:
(2.1 cm)(215 km /cm) = 451 km
(1.3 cm)(431 km/cm) = 560 km
Area = 451 x 560 km = 252,560 sq km
The sum of areas A, B, and C is 2,268,000 sq km
Since the widths of A and B are about the same and Greenland is wider than the average, an estimate of the width is 1200 km.
Second Method:
Since each square on the close up of Greenland has to be 10° and the distortion is only in the horizontal direction, divide Greenland into long skinny rectangles that are 1° tall and estimate the number of degrees wide. Since the southern tip of Greenland appears to touch the 60° latitude line, the center of each rectangle will be on half degree lines. The table below gives the latitude of the center line and the estimated degrees at that latitude.
| Lati-tude |
Deg. wide |
Width in km |
Area sq km |
If the radius of the earth is 6670 km there must be 2_ 6670 km for every 360° or 41910km/360° = 116.4 km/degree. Since there is no distortion vertically, Greenland must be (24°)(116.4 km/degree) approximately 2800 km tall. Also each increment in the first column is 1° which means the rectangles are 116.4 km tall. Since the radius of the circles at the different latitudes is a percentage of the earth radius and therefore so is the circumference and that percentage is exactly the cosine of the latitude, the length of the rectangles can be found by the formula: cos(latitude) x 116.4km/degree x length of the rectangle in degrees. Entering that in the list in the calculator, the third list represents the widths Greenland for each rectangle. From that list appears that Greenland is about 1250 km wide at the widest point. To find the area of each rectangle, column three is multiplied by 116.4 km/degree giving column four. For the total area column four is summed yielding an area of: 2,284,000 sq km. |
| 60.5 |
2 |
115 |
13344 | |
| 61.5 |
6 |
333 |
38790 | |
| 62.5 |
8 |
430 |
50050 | |
| 63.5 |
9 |
467 |
54410 | |
| 64.5 |
11 |
551 |
64163 | |
| 65.5 |
14 |
676 |
78661 | |
| 66.5 |
18 |
835 |
97247 | |
| 67.5 |
19 |
846 |
98514 | |
| 68.5 |
24 |
1024 |
119177 | |
| 69.5 |
26 |
1069 |
123369 | |
| 70.5 |
32 |
1243 |
144728 | |
| 71.5 |
26 |
960 |
111778 | |
| 72.5 |
28 |
980 |
114079 | |
| 73.5 |
32 |
1058 |
123140 | |
| 74.5 |
36 |
1120 |
130349 | |
| 75.5 |
37 |
1078 |
125518 | |
| 76.5 |
40 |
1087 |
126518 | |
| 77.5 |
49 |
1235 |
143694 | |
| 78.5 |
50 |
1160 |
135061 | |
| 79.5 |
46 |
976 |
113579 | |
| 80.5 |
47 |
903 |
105103 | |
| 81.5 |
49 |
843 |
98131 | |
| 82.5 |
32 |
486 |
56592 | |
| 83.5 |
12 |
158 |
18405 |
Actual values:
Width is about 1250 km and the height is about 2800 km. The area is 2,175,600 sq km