. The position of two towns X and Y are given to the nearest degree as X(45° N, 10° W) and Y (45 N°, 70° W).
Find
70-10 =60
(a) The distance between the two towns in
(i) Kilometers ( take the radius of the earth as 6370)
60/360 *2*3.142*6370*cos(45)
= 4717.47km
(ii) Nautical miles ( take 1 nautical mile to be 1.85 km)
(b) The local time at X when the local time at Y is 2.00 p.m.
I can't seem to figure out your bearing measures for where your two towns are. The question does not seem to use standard direction measures. If you can repost them we can solve them.
To find the local time at X when the local time at Y is 2:00 p.m., we need to consider the time difference between the two towns using their longitudinal coordinates.
The longitude difference between the two towns is 70° W - 10° W = 60°.
Since the Earth rotates 360° in 24 hours, we can calculate the time difference for each degree of longitude as 24 hours / 360° = 4 minutes per degree.
Therefore, the time difference between the two towns is 60° * 4 minutes = 240 minutes or 4 hours.
If the local time at Y is 2:00 p.m., then the local time at X would be 2:00 p.m. - 4 hours = 10:00 a.m.
To find the distance between two towns X and Y, given their positions, you can use the Haversine formula or the Spherical Law of Cosines. Let's use the Haversine formula in this case:
1) Convert the coordinates of the towns from degrees to radians.
X: (45° N, 10° W) => (45° * π/180, -10° * π/180)
Y: (45° N, 70° W) => (45° * π/180, -70° * π/180)
2) Calculate the central angle between the two towns using the longitude difference:
Δλ = (-70° - (-10°)) * π/180 = -60° * π/180
3) Calculate the distance between the two towns using the Haversine formula:
Distance = 2 * radius of Earth * arcsin(sqrt(sin²(Δφ/2) + cos(φ1) * cos(φ2) * sin²(Δλ/2)))
Using the given radius of the Earth as 6370 km, and the coordinates above:
Distance = 2 * 6370 * arcsin(sqrt(sin²((45° * π/180 - 45° * π/180)/2) + cos(45° * π/180) * cos(45° * π/180) * sin²((-60° * π/180)/2)))
Simplifying the above equation will give you the distance in kilometers.
(i) Kilometers:
The given coordinates already cover the latitude and longitude components required for the Haversine formula. Calculate the above equation using the given values to find the distance between the two towns in kilometers.
(ii) Nautical miles:
Multiply the distance in kilometers by the conversion factor, which is 1 nautical mile = 1.85 km, to convert the distance into nautical miles.
To find the local time at town X when the local time at town Y is 2.00 p.m., you need to consider the time difference between the two towns based on their longitudes.
1) Calculate the longitude difference between town X and town Y: -70° - (-10°) = -60°.
2) Each degree of longitude corresponds to 4 minutes of time difference. So, multiply the absolute value of the longitude difference by 4 to find the time difference between the two towns in minutes.
3) Add or subtract the time difference from the given time at town Y (2.00 p.m.) to find the local time at town X. Since the longitude of town Y is greater than town X, you need to subtract the time difference in this case.