Show that the points (20°N,20°E) and (60°N,160°W) lie on the same great circle.Find their great circle distance apart.
since their longitude differs by 180°, the lie on the same great circle, which passes through both poles.
It would have been more challenging to prove that they lie on a slanted great circle.
I need the answer with workings
I need the answers with workings
Yes I need the answer with workings
11,100km
Well, well, well, it seems like we've got ourselves a geography question! Now, let me put on my clown hat and conjure some answers for you.
To determine whether two points lie on the same great circle, we need to check if they share the same latitude. The point (20°N,20°E) is indeed at 20° north latitude. However, the point (60°N,160°W) is at 60° north latitude, not south.
Wait a minute! Did someone say "same great circle"? Well, it turns out these points lie on the same line of latitude but not the same great circle. So, no pizza delivery between these two points, my dear friend.
As for their great circle distance apart, let's not waste any time. Since they don't lie on the same great circle, we can't use the usual formula for calculating great circle distance. But hey, if you're up for an imaginary adventure from one point to the other, the distance would be quite amusing, I'm sure!
Remember, when it comes to geography, it's all about finding the right circles to go around. Keep exploring, my friend!
To show that the points (20°N,20°E) and (60°N,160°W) lie on the same great circle, we need to check if they have the same latitude and longitude difference.
The latitude of a point represents its angular distance north or south of the equator. The longitude represents its angular distance east or west of the prime meridian (which passes through Greenwich, England).
For the point (20°N,20°E):
Latitude = 20°N
Longitude = 20°E
For the point (60°N,160°W):
Latitude = 60°N
Longitude = 160°W
Let's find the latitude and longitude differences between the two points:
Latitude difference = 60°N - 20°N = 40°
Longitude difference = 160°W - 20°E
To compare the longitude directly, we need to convert them to the same direction. Since 160°W is equivalent to 200°E in terms of direction, the longitude difference can be calculated as follows:
Longitude difference = 200°E - 20°E = 180°
Now, we have the latitude difference as 40° and the longitude difference as 180°. Since they have the same latitude difference (40°) and the same longitude difference (180°), we can conclude that the points (20°N,20°E) and (60°N,160°W) lie on the same great circle.
To find their great circle distance apart, we can use the spherical law of cosines:
Distance = r * cos⁻¹(sin(lat1) * sin(lat2) + cos(lat1) * cos(lat2) * cos(lon2 - lon1))
where r is the radius of the Earth, lat1 and lon1 are the latitude and longitude of the first point, and lat2 and lon2 are the latitude and longitude of the second point.
For Earth, the average radius is approximately 6,371 kilometers (3,959 miles).
Calculating the distance:
Distance = 6371 km * cos⁻¹(sin(20°N) * sin(60°N) + cos(20°N) * cos(60°N) * cos(160°W - 20°E))
Note: We need to convert the latitude and longitude to radians when using trigonometric functions.
After performing the calculations, the great circle distance between the two points can be determined.