  Question

Suppose that the earth is a sphere of radius $$6400$$ kilometers. The height from the earth's surface from where exactly a fourth of the earth's surface is visible, is:

A
3200km  B
32002km  C
32003km  D
6400km  Solution

The correct option is D $$6400 km$$Let the height above the Earth be $$h$$ as shown in the figure.Only the spherical cap bounded by $$A$$ and $$A'$$ will be visible from that height. The total surface area of Earth is $$4\pi R^2$$.If $$\frac{1}{4}$$ area of Earth is visible from $$B$$, then, area of spherical cap $$AA'$$ is $$\pi R^2$$We know that, area of spherical cap, that subtends an semi-apex angle $$\theta$$ is $$2\pi R^2 (1-\cos\theta)$$Here, $$\theta$$ is $$\angle AOB$$From the Right triangle $$\triangle OAB$$, $$\cos \theta = \frac{OA}{OB} = \frac{R}{R+h}$$And, $$2\pi R^2(1-\cos\theta)=\pi R^2 \Rightarrow \cos\theta = \frac{1}{2}$$Therefore, $$\frac{R}{R+h} = \frac{1}{2}$$$$h = R$$ Physics

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