Question

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 \(\_\_\_\_\).

• A. \((2 - \sqrt{3}) \pi R^2\)
• B. \(\frac{(\sqrt{2} - 1) \pi R^2}{2}\)
• C. \(\frac{2 \pi R^2}{3}\)
• D. \(\frac{(2 + \sqrt{3}) \pi R^2}{8\sqrt{2}}\)

Hint:

To calculate the surface area of a spherical cap, use the formula \(A = 2\pi R^2 (1 - \cos \theta)\), ensuring \(\theta\) is measured from the pole.

Answer 1

The Correct Option is A

Step 1: Recall the formula for the surface area of a spherical cap.
The surface area of a spherical cap is given by: \[ A = 2\pi R^2 (1 - \cos \theta), \] where \(R\) is the radius of the sphere, and \(\theta\) is the angle measured from the pole. Step 2: Determine \(\cos \theta\).
The latitude \(60^\circ N\) corresponds to an angle of \(\theta = 30^\circ\) from the pole. The cosine of this angle is: \[ \cos 30^\circ = \frac{\sqrt{3}}{2}. \] Step 3: Substitute into the surface area formula.
Substitute \(R\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) into the formula: \[ A = 2\pi R^2 \left(1 - \frac{\sqrt{3}}{2}\right). \] Simplify: \[ A = (2 - \sqrt{3}) \pi R^2. \] Final Answer: \[\boxed{{(1) } (2 - \sqrt{3}) \pi R^2}\]