GATE EC 2024 Set 1 — Question 36
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Engineering Mathematics → Calculus → Double & Triple Integrals
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Question
Consider the Earth to be a perfect sphere of radius . Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is ________.
Correct answer
(A) (2-√(3))π R²
Solution
The problem asks for the surface area of a spherical cap. The region enclosed by the 60°N latitude circle that contains the north pole is a spherical cap.The formula for the surface area of a spherical cap is , where is the radius of the sphere and is the polar angle (the angle measured from the north pole to the edge of the cap).The given latitude is 60°N. The latitude is the angle measured from the equator. The polar angle is related to the latitude by .For 60°N latitude, .
So, the polar angle .Now, we need to find .
.Substitute this value into the formula for the surface area of the spherical cap:
Comparing this with the given options, option (A) matches the calculated area.The final answer is .
So, the polar angle .Now, we need to find .
.Substitute this value into the formula for the surface area of the spherical cap:
Comparing this with the given options, option (A) matches the calculated area.The final answer is .
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