A and B are points on the parallel of latitude 58.4N their longitudes being 148W and 32E respectively.What is their distance apart (a)along the parallel of latitude (b)along a great circle
earth radius is about 6,378.* 10^3 meters or about 3,440 nautical miles. You do not say which you want so I will use nautical miles (60 nm = 1 degrees on a great circle)
so what is the radius of a circle parallel to the equator at 58.4 N?
r = R cos (latitude angle) = 3,440 * cos 58.4 = 1802.5 nm
so our distance along that 58.4 parallel is
1802.5 * 2 pi * (148+32)/360
but note that 148+32 = 180, so halfway around :)
so 1802.5 * 3.14 = 5660 nautical miles
Now the spherical geometry thing
we need to draw lines from the center of the earth to these two points and find the angle between those lines at the center.
BUT
they are 180 degrees apart in longitude
so if you look up at them from the center of a transparent earth the great circle goes right over the north pole :)
therefore we go (90 - 58.4) up and the same down
so 2 * 31.6 = 63.2 degrees around great circle
63.2 * 60 nm /degree = 3,792 nautical miles
Put your winter boots on and do it over the pole, not around that constant latitude.
This by the way is why ships (like the Titanic) between Europe and N America end up traveling much further North than their starting point or their finish point.
And do not try it tonight. The wind is shaking the house and I can hear the waves crashing on the breakwater down at the Cove.
To determine the distance between two points on the Earth's surface, we can use the Haversine formula. However, before we can apply the formula, we need to convert the given longitudes into a common format.
a) To find the distance between the two points along the parallel of latitude, we can use the formula:
Distance = (Longitude A - Longitude B) * (111 km/degree)
First, we need to convert the longitudes into a common format. Since the longitudes for point A and B are given as 148W and 32E respectively, we should convert them to degrees format.
To convert a longitude given in West (W) or East (E) to degrees format, we can use the following conversions:
For West longitude, the value is negative:
-148° = -1 * 148° = -148°
For East longitude, the value is positive:
32° = 32°
Now we have the longitudes in degrees format: -148° and 32°. We can calculate the distance along the parallel of latitude.
Distance = (-148° - 32°) * (111 km/degree)
Distance = (-180°) * (111 km/degree)
Distance = -19980 km
So, the distance along the parallel of latitude is -19980 km.
b) To find the distance between the two points along a great circle, we use the Haversine formula:
Distance = 2 * RADIUS * arcsin(sqrt(sin^2((Latitude A - Latitude B)/2) + cos(Latitude A) * cos(Latitude B) * sin^2((Longitude A - Longitude B)/2)))
Where RADIUS is the radius of the Earth (approximately 6371 km).
For the latitude, both points are given as 58.4N, so we can directly use this value in the formula.
For the longitude, we already converted them to degrees format: -148° and 32°.
Now, we can calculate the distance along the great circle.
Distance = 2 * 6371 km * arcsin(sqrt(sin^2((58.4N - 58.4N)/2) + cos(58.4N) * cos(58.4N) * sin^2((-148° - 32°)/2)))
Distance = 2 * 6371 km * arcsin(sqrt(sin^2(0) + cos(58.4N) * cos(58.4N) * sin^2(-90°)))
Distance = 2 * 6371 km * arcsin(sqrt(0 + cos(58.4N) * cos(58.4N) * 1))
Distance = 2 * 6371 km * arcsin(sqrt(cos(58.4N) * cos(58.4N)))
To evaluate this expression, we need the cosine of 58.4N (58.4 degrees). Using a scientific calculator or trigonometric table, we find that cos(58.4) is approximately 0.53027.
Distance = 2 * 6371 km * arcsin(sqrt(0.53027 * 0.53027))
Distance = 2 * 6371 km * arcsin(0.53027)
Distance = 2 * 6371 km * 0.55472
Distance = 7016 km
So, the distance along a great circle is approximately 7016 km.