All Solutions - Great Circles, Nautical Miles - May 27-31, 1996
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Correct solutions were submitted by:
Akiba Hebrew Academy, Merion, Pennsylvania David Love, Grade 11 Franklin County High School, Rocky Mount, Virginia Rita Beckner, Molly Dickerson and Melissa Foster, Grade 9 Granada High School, Livermore, California Ethan Castor, Mike Sue, Zac Crawford, and Neil Tucker, Grade 10 J.P. Taravella High School, Broward County, Florida Brent Tworetzky, Grade 9 Livermore HS, Livermore, California Brooke Freeman and Danny Farrell, Grades 10 & 11 Mt. St. Josephs, Flourtown, Pennsylvania Jackie Benn and Shannon Firth, Grade 9 Newport High School, Bellevue, Washington Adam Morley, Grade 8 Waluga Jr. High School, Lake Oswego, Oregon Adam Wright, Grade 8 Brian Gordon, Dartmouth '92, Wethersfield, Connecticut
Rita Beckner grade 9 Franklin County High School Rocky Mount, Virginia The nautical mile measuring 1.852 kilometers is a good compromise. First of all, the circumference at the equator is 40074.16 (pi times diameter, 12756). This divided by 360 to make degrees is 111.32. Then one must divide by 60 to find the measure of a minute. Calculated, this is 1.86. The circumference between the poles is 39942.209. Then one must divide by 360, then 60, totalling to be 1.85. Having both these circumferences, one must next add and divide by 2 (average). The calculation of 1.852 kilometers, used as a nautical mile, proves to be a good compromise. ********************************************** From: Brent Tworetzky Grade: 9 School: J.P. Taravella High School (Broward County, Fla.) This question is just inserting the radius into a formula. That formula is 1 * (2)(pi)(radius) {one minute * the diameter of the great circle} --------------- {divided by} (60)(360) {total minutes * total degrees} thus, by substituting 6378 and 6357, we get approximately 1.855 and 1.849, respectively. 1.852 is a good approximation, as it is the arithmetic mean of our two solutions. ********************************************** Jackie Benn and Shannon Firth Mt. St. Josephs Grade 9 Well, we know that the equator's radius is 6378 km and that the radius of the circle that goes through the poles is 6357 km. So the first step here is to find the circumference of each circle, we can do this by plugging in the radii to the formula for circumference which is 2piR. When we do this, we get for the circumferences: Equator- 40,074.156 km and for the poles- 39,942.209 km. Okay, now the second step to the problem: Remembering what you had said, that one minute of latitude was a nautical mile, and knowing that 1degree of a circle is 1/360 and that a minute is 1/60 of an hour, we multiplied 1/360 by 1/60 and we got 1/21600. Now to find 1 nautical mile of the equator circle and the circle of the poles, we multiplied the two different circumferences by 1/21600. So this is what it looked like: nautical mile for the equator- 40,074.156 km/21600 this equaled 1.855285 km for a nautical mile traveling across or latitude wise or perpendicular to the equator. nautical mile for the circle of the poles would look like : 39,942.209km/21600 and this would equal 1.8491763 km. I think 1.852 would be a pretty good guess for a nautical mile because when I averaged the two numbers I got for the two different nautical miles, I got 1.8522307. Very close! Also, knowing that people don't always travel completely in the directions of latitude or longitude is another reason why I averaged the two numbers. ********************************************** From: Adam Morley Grade: 8 School: Newport High School There are 360 degrees around the earth and there are 60 minutes in every degree. So there are 21600 minutes around the earth. If the radius of the great circle around the earth at the poles is 6357km then the diameter is 12714km. Since the way to find the circumference of a circle is d*pi the circumference around is 39942.2089977km. Then you divide that by 21600 (because that is the distance around in minutes) and you get 1.84917634249km per minute. Then we take the radius 6378km for the equator and double it for the diameter to get 12756 and multiply by pi to get 40074.1558892km around at the equator. Then you divide by 21600 to get the distance around in minutes and get 1.8552849948km per minute. You average those two together and get 1.8522306686km per minute average. That rounds to 1.852 which is what a nm is. Yes, I do think it is close enough. On another note, I really enjoyed when you came and taught us about spherical geometry. Oh, yes make sure you warn us before you come back so we can get ready for the wrath of God. The first time you came we had floods. The second we had an earthquake. What next? Just enough time so we can get out of Dodge. Thanx! ********************************************** From: Adam Wright Grade: 8 School: Waluga Jr. High School To solve this problem, I used the formula for circumference, or (pi)d. Since we know the radii of the earth, we need to use the formula 2r(pi). A nautical mile is the ratio of the circumference (2 pi R) and the number of minutes in the earth. Since there are 360 degrees in the earth, and there are 60 minutes in a degree, we know that the following formula can be used to figure out a good length for a nautical mile with any radius: 2(pi)r ------ 21600 Using this formula, we know that a good length for a mile around the equator is 1.855, and around the poles is 1.849. If we figure out the average of these two numbers we get 1.852. Since the estimate for a nautical mile is 1.852, it is accurate. So, yes, this is a good estimate. ********************************************** Molly Dickerson and Melissa Foster grade 9 Franklin County High School Rocky Mount, VA First we figured out that each of the great circles has 360 degrees in them and each degree has 60 minutes. So that would be 21600 nautical miles. Then we figured out the circumference of each circle by multiplying the radius by two and then multiplying it by pi. The circle around the poles has a circumference of 39942.209 and the one around the equator has a circumference of 40074.15589. So we divided each circumference by 21600 nautical miles and came up with 1.8491 for each nautical mile around the poles and 1.8552 around the equator. So we decided that 1.852 was a fair number for each nautical mile because it's an average of the other two. ********************************************** From: Brooke Freeman and Danny Farrell Grade: 10 & 11 School: Livermore High School 1. First we figured out the circumferences of each "Great Circle" using 3.14 as pi. 2. C1 = 6378 x 3.14 x2 = 40053.84 C2 = 6357 x 3.14 x 2 = 39921.96 3. Then you divide it by 360 to get C1=111.26 km/degree and C2 = 110.89 km/degree 4. Then divde each by 60 to get the nautical miles C1 = 1.854 km/min and C2 = 1.848 km/min 5. Then find the average 1.848+1.854 = 1.851 nautical miles 6. So yeah you are right to use 1.852 because you were very close. ********************************************** Ethan Castor 10th Granada High School Livermore,Ca I think the figure of 1.852 km/nautical mile is a perfectly good compromise. How I got to my conclusion is ... ((6378*2*pi)/360)/60 = 1.855 Equatorial ((6357*2*pi)/360)/60 = 1.849 Polar 1.852 is halfway between the two numbers. ********************************************** David Love Akiba Hebrew Academy 11th grade To do this problem you want to compare the lengths for both circles. In order to get their lengths, you must find the circumferences of each one's circle. The great circle around the equator has a circumference of 2x times 6378, or 40074.15. The great circle around the poles has a circumference of 2x times 6357, or 39942.21. To find the central angle, you know it is one minute of one degree, or 1/21600 of the circle. If for each circle you divide the circumference by(1 over what part of the circle is represented), you get 40074.15/21600 = 1.85528 for the equator, and 39942.21/21600 = 1.84918 for the poles. If you find their average (by adding them together and dividing the result by two) you get 1.85223. Therefore you can conclude that 1.852 kilometers is a good compromise, as it is halfway between the two. ********************************************** From: Mike Sue, Zac Crawford, and Neil Tucker Grade: 10 School: Granada 1 min of equator arc = 1 min/360 degree x 1 degree/60 minutes x 2 x Pi x 6378 km = 1.855 km 1 min of polar arc = 1 min/360 degree x 1 degree/60 minutes x 2 x Pi x 6357 km = 1.849 km 1 nautical mile = 1.852 km = (1.855 x 1.849)/2 = 1.852 km 1.852 km = the average of 1 min of an equatorial arc and a polar arc 1.852 km is a good compromise because it's the average. ********************************************** From: Brian Gordon Grade: 1992 School: Dartmouth One minute is 1/60 * 1/360 of the circumference of each of these circles. So for each, I multiplied 2 * pi* radius, and then divided by 21,600. Here's what I got: equator: 6378 * 2 * pi / 21600 = 1.855 km poles: 6357 * 2 * pi / 21600 = 1.849 km Sounds like 1.852 km is a pretty good compromise to me. --bri
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