P and Q are two towns on the earth surface on latitude 56 degrees N.their longitudes are 25 degrees E and 95 degrees E respectively. Find the distance between PQ along their parallel of latitude, correct to the nearest km.
To find the distance between towns P and Q along their parallel of latitude, you can use the haversine formula. The haversine formula allows us to find the distance between two points on the Earth's surface using their latitude and longitude coordinates.
First, let's convert the given latitudes and longitudes from degrees to radians:
Latitude of P (ϕ₁) = 56 degrees N = 56° × (π/180) = 0.97738438111 radians
Latitude of Q (ϕ₂) = 56 degrees N = 56° × (π/180) = 0.97738438111 radians
Longitude of P (λ₁) = 25 degrees E = 25° × (π/180) = 0.436332313 radians
Longitude of Q (λ₂) = 95 degrees E = 95° × (π/180) = 1.658062789 radians
Next, we can use the haversine formula to calculate the distance (d) between two points (P and Q) on the Earth's surface:
hav(d) = hav(ϕ₂ - ϕ₁) + cos(ϕ₁) × cos(ϕ₂) × hav(λ₂ - λ₁)
Where hav(x) = sin²(x/2).
Now, we can substitute the values into the formula:
hav(d) = hav(0.97738438111 - 0.97738438111) + cos(0.97738438111) × cos(0.97738438111) × hav(1.658062789 - 0.436332313)
Simplifying the equation:
hav(d) = hav(0) + cos(0.97738438111) × cos(0.97738438111) × hav(1.22173047567)
Using an online calculator (or a scientific calculator that supports haversine functions), we find:
hav(d) ≈ 0.03382882051
Now, we need to find the inverse haversine (inverse haversine(x) = 2 × arcsin(√(x))).
Using the inverse haversine:
d ≈ 2 × arcsin(√(0.03382882051))
d ≈ 0.116573413235 radians
Finally, to convert the distance from radians to kilometers, we can use the Earth's radius. Taking the average radius of the Earth as 6,371 kilometers:
Distance (D) ≈ 0.116573413235 × 6,371 km
D ≈ 742.011526 km
Therefore, the distance between towns P and Q along their parallel of latitude is approximately 742 kilometers.
Distance pq = 70 /360 x 2πr where r=Rcos 56° =7/36*2*22/7*6400*0.5592 =1/9*11*6400*0.5592.
=4374.13km=4374km to the nearest km.