Losing Your Bearings
1. Begin journey at 40 degrees N by 77 degrees W.
2. Travel 4200 miles with a compass bearing of 130 degrees.
3. Travel 2800 miles with a compass bearing of 180 degrees.
4. Travel 5250 miles with a compass bearing of 90 degrees.
5. Travel 1750 miles with a compass bearing of 0 degrees.
6. Where is your location using latitude and longitude coordinates?
1 degree latitude = 70 miles
I know how to do this, however, your 1 degree latitude is just wrong. It could be that at some latitude, say the starting point, however, as soon as you go N or S it changes. Steps 3, and 5 is a considerable change in latitude, so the 1 degree = fixed value is wrong.
As one goes N or S, the conversion of miles to latitude difference changes. So even in step1, and step 4, it has to be considered. In step 4, you are not at the starting point anymore.
So how does one work this? The easy way is to plot it on a globe. The wrong way to work it is to plot it on a mercator projection (flat map).
The complicated way to solve it is to use spherical trignometry, which I doubt you have been introduced to.
http://star-www.st-and.ac.uk/~fv/webnotes/chapter2.htm
As you can see, the problem is that as the problem states the travel as straight paths, the real path traveled in each step was a curve on the surface of the Earth, not a straight line. Any attempt to measure it as a straight line is wrong. When the distances are large, as in this problem, the error is very large.
Plot this on a globe.
A reminder: Due to the curvature, step one 4200miles at a compass bearing 130 will make a spiral upwards as it is drawn on a sphere. You will have to be careful drawing the line segments to make it a constant angle with Longitude lines.
Wouldn't life be simple if you could use a flat map and ignore the fact that it is totally a wrong representation of Earth's surface?
You are absolutely correct that using a flat map, like a Mercator projection, can result in significant inaccuracies when trying to measure distances and angles on the Earth's surface. The Earth is a three-dimensional object, so representing it on a flat surface will inevitably involve distortions.
To accurately solve the problem at hand, plotting the points and distances on a globe or a spherical map would be the most accurate method. However, since you mentioned that you know how to solve this problem, I'll provide you with the steps to do it regardless of the inaccuracies that come from using a flat map.
To determine your location using latitude and longitude coordinates, you need to follow these steps:
1. Start at 40 degrees N by 77 degrees W. This is your initial position.
2. Travel 4200 miles with a compass bearing of 130 degrees. To start from your initial position, draw a straight line at a 130-degree angle from north and measure 4200 miles along that line. This will give you your new position on the flat map.
3. Travel 2800 miles with a compass bearing of 180 degrees. From your new position, draw a straight line at a 180-degree angle from north and measure 2800 miles along that line. This will give you another new position on the flat map.
4. Travel 5250 miles with a compass bearing of 90 degrees. From your second new position, draw a straight line at a 90-degree angle from north and measure 5250 miles along that line. This will give you a third new position on the flat map.
5. Travel 1750 miles with a compass bearing of 0 degrees. Finally, from your third new position, draw a straight line at a 0-degree angle from north and measure 1750 miles along that line. This will give you your final position on the flat map.
Keep in mind that these measurements are only approximate due to the distortions caused by using a flat map. To obtain a more accurate location, using a globe or spherical map would be necessary.
If you have access to tools like Google Earth or other online mapping resources, you can input the latitude and longitude coordinates of your starting point and then use the measuring tools to determine the distances and positions more accurately.