Two town P and Q are on the parallel of latitude 46° N .the longitude of P is 130°W and that of town Q is 103°W.a third town R also on latitude 46°N on longitude 23°E.calculate
(A).the length of the parallel of latitude 46°N to the nearest 100km
(B).the distance between Q and R measure along the parallel of latitude to the nearest 10km(take π=3.142and radius of the Earth =6400km)
A. using a planar section through the Earth from pole to pole
... construct a right triangle using the center of the Earth, the equatorial plane, and the line of 46ºN
... the hypotenuse is the radius of the Earth from the center to 46ºN
... the side lying in the equatorial plane is the radius of the 46ºN parallel
... the length of the parallel is 2π times the radius
B. Q and R are 126º apart on the 360º length of the 46ºN parallel
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To solve this problem, we need to apply some basic principles of geography and trigonometry. Let's break down the problem step by step:
(A) Calculating the Length of the Parallel of Latitude 46° N:
To calculate the length of a parallel of latitude, we need to know the difference in longitudes between the two towns. In this case, the longitude of town P is given as 130°W, and the longitude of town Q is given as 103°W. To find the difference in longitudes:
1. Convert the longitudes to positive values: Since the longitudes are given as west, we need to convert them to positive values by adding 360°.
Longitude of P (130°W) = 130° + 360° = 490°
Longitude of Q (103°W) = 103° + 360° = 463°
2. Find the difference in longitudes: Subtract the longitude of town Q from the longitude of town P.
Difference in longitudes = Longitude of P - Longitude of Q
= 490° - 463°
= 27°
3. Calculate the length of the parallel of latitude: The length of a parallel of latitude can be calculated using the formula:
Length of parallel of latitude = (Difference in longitudes / 360°) * Circumference of the Earth
Given: Difference in longitudes = 27°, Circumference of the Earth = 2 * π * Radius of the Earth
Radius of the Earth is given as 6400 km, so substituting the values:
Length of parallel of latitude = (27° / 360°) * (2 * 3.142 * 6400 km)
= (27/360) * (2 * 3.142 * 6400 km)
≈ 481 km
Therefore, the length of the parallel of latitude 46° N is approximately 481 km to the nearest 100 km.
(B) Calculating the Distance between Q and R along the Parallel of Latitude:
To calculate the distance between two towns along a parallel of latitude, we can use the arc length formula. Given the radius of the Earth as 6400 km and the longitudes of towns Q and R, we can calculate the distance:
1. Convert the longitudes of towns R and Q to positive values:
Longitude of R (23°E) = 23°
Longitude of Q (103°W) = 103° + 360° = 463°
2. Find the difference in longitudes:
Difference in longitudes = Longitude of Q - Longitude of R
= 463° - 23°
= 440°
3. Convert the difference in longitudes to arc length: The arc length can be calculated using the formula:
Arc length = (Difference in longitudes / 360°) * Circumference of the Earth
Given: Difference in longitudes = 440°, Circumference of the Earth = 2 * π * Radius of the Earth
Substituting the values:
Arc length = (440° / 360°) * (2 * 3.142 * 6400 km)
= (440/360) * (2 * 3.142 * 6400 km)
≈ 9965 km
Therefore, the distance between towns Q and R, measured along the parallel of latitude 46° N, is approximately 9965 km to the nearest 10 km.