Two cities are due north of each other. One city has a latitude of 32¢ª45'12" and the second city has a latitude of 51¢ª12'45". Assuming that the earth is a sphere with a radius of 4000 miles, how far apart are the two cities?
Same longitude, so you can just use an arc distance: radius*(difference in latitudes).
Once you do that, calculate it with a better computer model, with real Earth radius. Choose any longitude you wish, as long as they are the same.
http://www.movable-type.co.uk/scripts/latlong.html
50 72 45 (the 72 is 60+12)
32 45 12
----------subtract
18 27 33
now being a navigator, I happen to know that a minute of latitude is a nautical mile so
I would say
18*60 + 27 + 33/60
= 1107.5 NAUTICAL miles
However do not tell you math teacher that :)
we have 1107.5 Minutes of arc
that is
1107.5/(360*60) = .051275 of the circumference of the earth
which is 2 pi r = 25132.7 miles
1289 miles
Those are LAND miles, not used in navigation.
To find the distance between the two cities, we can use the formula for the distance on a sphere between two points given their latitudes and longitudes, known as the haversine formula.
The haversine formula is given by:
d = 2r * arcsin(sqrt((sin²((lat2-lat1)/2)) + cos(lat1) * cos(lat2) * sin²((lon2-lon1)/2))))
Where:
- d is the distance between the two points on the sphere
- r is the radius of the sphere
- lat1 and lat2 are the latitudes of the two points in radians
- lon1 and lon2 are the longitudes of the two points in radians
First, let's convert the given latitudes to radians:
lat1 = 32°45'12" = 32 + (45/60) + (12/3600) degrees
= 32.75333 degrees
= 32.75333 * (π/180) radians
lat2 = 51°12'45" = 51 + (12/60) + (45/3600) degrees
= 51.2125 degrees
= 51.2125 * (π/180) radians
Next, let's calculate the distance using the haversine formula:
d = 2 * 4000 * arcsin(sqrt((sin²((51.2125 - 32.75333)/2)) + cos(32.75333) * cos(51.2125) * sin²((0-0)/2))))
Note: The longitudes are not given, but since the two cities are due north of each other, we can assume their longitudes to be the same and therefore set lon1 = lon2 = 0.
Simplifying the equation further:
d = 2 * 4000 * arcsin(sqrt((sin²((51.2125 - 32.75333)/2)) + cos(32.75333) * cos(51.2125) * sin²(0/2))))
= 2 * 4000 * arcsin(sqrt((sin²(9.45917)) + cos(32.75333) * cos(51.2125) * sin²(0))))
= 2 * 4000 * arcsin(sqrt((0.923048722) + (0.828569761) * (1) * (0))))
= 2 * 4000 * arcsin(sqrt(0.923048722))
= 2 * 4000 * arcsin(0.960487605)
= 2 * 4000 * (1.229758378)
= 2 * 4000 * 1.229758378
= 2 * 4919.00677991
= 9838.01355983 miles
Therefore, the two cities are approximately 9838.01355983 miles apart.
To find the distance between the two cities, we can use the Haversine formula. The Haversine formula calculates the distance between two points on a sphere given their latitudes and longitudes.
First, we need to convert the latitudes from degrees, minutes, and seconds to decimal degrees.
For the first city:
32°45'12" = 32 + 45/60 + 12/3600 = 32.75333333°
For the second city:
51°12'45" = 51 + 12/60 + 45/3600 = 51.2125°
Now, we can calculate the angular distance (in radians) between the two cities using the Haversine formula:
Δσ = acos(sin(Φ1) * sin(Φ2) + cos(Φ1) * cos(Φ2) * cos(Δλ))
Where:
- Δσ is the angular distance
- Φ1 and Φ2 are the latitudes of the two cities (in radians)
- Δλ is the difference in longitudes between the two cities (in radians)
Next, we need to convert the latitudes from decimal degrees to radians:
Φ1 = 32.75333333° * π/180 = 0.57275504 radians
Φ2 = 51.2125° * π/180 = 0.89324341 radians
Since the cities are due north of each other, their longitudes are the same, so Δλ = 0.
Now, we can plug in the values into the formula:
Δσ = acos(sin(0.57275504) * sin(0.89324341) + cos(0.57275504) * cos(0.89324341) * cos(0))
Calculating this expression gives us the angular distance between the two cities.
Finally, we can find the actual distance by multiplying the angular distance by the radius of the Earth:
Distance = Δσ * Radius
Plugging in the values for Δσ (in radians) and the Earth's radius (4000 miles), we can calculate the final result which gives us the distance between the two cities.