What part of the planet is inside the horizon of the prince?

Question

http://upload.wikimedia.org/wikipedia/en/0/05/Littleprince.JPG

His height is 4 feet and radius of the planet is 6 feet.

What part of planet surface

Answer 1

Tony the Dad started out OK, but made a mistake calculating the surface area of the visible region. It's actually 3/10 of the planet's total area.

First, draw a 2D diagram with the planet as a circle of radius R, and the prince as a line segment of height h.

Draw a line of length R+h, extending from the center C of the circle to a point H on top of the prince's head. That will (soon) be the hypotenuse of a right triangle.

Now draw a line segment from the top of the prince's head (H), to a point P tangent to the circle. Clearly point P is on the prince's horizon. This line segment is one of the legs of the right triangle we're drawing.

Finally, draw a line segment from circle's center (C) to P. That's the final leg.

Take the angle formed at C, and call that theta (θ). At this point, you should easily be able to figure out cosθ in terms of R and h; we'll use that later.

Now we want to figure out the portion of the sphere's surface area that is subtended by a "cone" whose vertex angle is 2θ (why?)

(This is where Tony the Dad went wrong.)

With calculus, you can show that this subtended area equals:

A = 2πR²(1–cosθ)

That's in square feet. If you want this as a fraction of the total sphere, divide it by the sphere's surface area (4πR²)

You now have all the formulas and all the numbers. Calculate!

Answer 2

d = √(2Rh + h²),
where R (radius of planet) and h (height of observer) are in the same units.

d = √(2·6·4 + 4²)
= √(48 + 16) = √64 = 8

So the horizon in any particular direction from the prince is 8 feet.

The area of the circle surrounding the prince 8 ft around is:
A = (8/6π)·4πr²
= (32/6)·36
= 6·32 = 192 ft²

The total surface area is
A = 4 πr² = 4·π·6² = 144π ft²

The ratio 192/(144π) gives .4244 or 42.44%