GATE 2024 EC – Question 36

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the $60^\circ$N latitude circle, that contains the north pole in its interior is ________.

$(2-\sqrt3)\pi R^2$
$\dfrac{(\sqrt2-1)\pi R^2}{2}$
$\dfrac{2\pi R^2}{3}$
$\dfrac{(2+\sqrt3)\pi R^2}{8\sqrt2}$

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Correct answer: (A) $(2-\sqrt3)\pi R^2$

Explanation

The 60°N circle is at a polar angle (colatitude) of $30^\circ$ from the north pole. The area of a spherical cap of angular radius $\theta$ is $2\pi R^2(1-\cos\theta)$. So the area is $2\pi R^2\left(1-\frac{\sqrt3}{2}\right)=(2-\sqrt3)\pi R^2$.