Answer:
Step-by-step explanation:
To solve this problem, we can use the concept of spherical geometry.
Given:
The distance between the two cities along the same north-south line is 2000 km.
The latitude of the northernmost city is 52 degrees N.
The radius of the Earth is approximately 6400 km.
Let's denote:
dd as the distance along the surface of the Earth between the two cities.
θθ as the angle formed at the center of the Earth between the two cities.
RR as the radius of the Earth.
ϕ1ϕ1 and ϕ2ϕ2 as the latitudes of the northernmost city and the other city, respectively.
We can use the formula for arc length on a sphere:
d=R⋅θd=R⋅θ
We know that the angle θθ is related to the difference in latitudes between the two cities. Since the cities lie on the same north-south line, the angle θθ is the difference in latitude between the two cities.
Given that the difference in latitude between the cities is 52∘−ϕ252∘−ϕ2, we need to convert this to radians:
θ=52∘−ϕ2180∘⋅πθ=180∘52∘−ϕ2⋅π
We also know that d=2000d=2000 km and R=6400R=6400 km.
Therefore, we have:
2000=6400⋅52∘−ϕ2180∘⋅π2000=6400⋅180∘52∘−ϕ2⋅π
Now, we can solve for ϕ2ϕ2:
20006400⋅π=52∘−ϕ2180∘6400⋅π2000=180∘52∘−ϕ2
13.2⋅π=52∘−ϕ2180∘3.2⋅π1=180∘52∘−ϕ2
13.2⋅π=52−ϕ21803.2⋅π1=18052−ϕ2
To solve for ϕ2ϕ2, we can cross multiply:
180=(52−ϕ2)⋅(3.2⋅π)180=(52−ϕ2)⋅(3.2⋅π)
180=52⋅(3.2⋅π)−ϕ2⋅(3.2⋅π)180=52⋅(3.2⋅π)−ϕ2⋅(3.2⋅π)
ϕ2⋅(3.2⋅π)=52⋅(3.2⋅π)−180ϕ2⋅(3.2⋅π)=52⋅(3.2⋅π)−180
ϕ2=52⋅(3.2⋅π)−1803.2⋅πϕ2=3.2⋅π52⋅(3.2⋅π)−180
Now, we can calculate ϕ2ϕ2:
ϕ2=52⋅3.2⋅π−1803.2⋅πϕ2=3.2⋅π52⋅3.2⋅π−180
ϕ2=166.4π−1803.2πϕ2=3.2π166.4π−180
ϕ2=166.4π3.2π−1803.2πϕ2=3.2π166.4π−3.2π180
ϕ2=52−1803.2πϕ2=52−3.2π180
ϕ2≈52−18010.08ϕ2≈52−10.08180
ϕ2≈52−17.857ϕ2≈52−17.857
ϕ2≈34.143ϕ2≈34.143
Rounding to the nearest integer, the latitude of the other city is approximately 34∘34∘ N.
