Ratios and Geometry

Date: 10/29/98 at 20:59:07
From: Kelly Blackwood
Subject: Geometry

An airplane flying at 33,000 feet has a visibility of 100 miles in any 
direction. What percent of the total land area to the horizon is 
visible? 

The diameter of the earth is 7930 miles. The answer that is given is 
20.1 percent. How did they arrive at this?

Date: 10/30/98 at 09:03:42
From: Doctor Rick
Subject: Re: Geometry

Hi, Kelly. The main part of this problem is to determine the distance 
to the horizon. Here is how to do it.

In the figure below, the circle is the earth, with center at point O. 
The radius OB is 7930/2 = 3965 miles. The airplane is at point A. The 
altitude is AC = 33000 feet = 6.25 miles (I've exaggerated the altitude 
a bit in the picture!) Line AB is the line of sight to the horizon. It 
is tangent to the surface of the earth, so angle ABO is a right angle.

                       A
                       |\
                       | \
                       |   \
                       |    \
                       |      \
                      C|       \
                    ******       \
              *        |       *  \
           *           |          * \
         *             |            *\
       *               |              * B
      *                |             / *
     *                 |          /     *
    *                  |       /         * 
   *                   |    /             *
   *                   | /                *
   *                   O                  * 
   *                                      *
    *                                    *
     *                                  *
      *                                *
       *                              *
         *                          *
           *                      *
              *                *
                    ******

You can figure the distance to the horizon, AB, by using the 
Pythagorean theorem, with OB = 3965 miles and OA = 3965 + 6.25 = 
3971.25 miles. It comes out to:

   sqrt(3971.25^2 - 3965^2) = 222.71 miles

Since the distance to the horizon is much less than the radius of the 
earth (unlike the figure), you can approximate the visible area by 
assuming that the earth is flat. Use Pythagoras again to get the 
distances on the surface:

         |\
         |   \    \        222.71
    6.25 |      \        \
         |    100  \            \
         |            \                \
         |_______________\____________________\
                99.8
                        222.62

The final step is to find the ratio of the areas of two circles: one 
with radius 99.8 miles, the other with radius 222.62 miles. This is 
easy. The ratio of the areas of two similar figures (like two circles) 
is just the square of the ratio of corresponding lengths (like the 
radii). So the ratio we want is:

   (99.8/222.62)^2 = 0.201 = 20.1%

If we had just used the line-of-sight distances (100 and 222.71 miles), 
we would be pretty close. It's just an approximation, anyway. You could 
figure the actual distance to the horizon along the ground by finding 
the angle AOB in radians and multiplying by OB. It's 222.48 miles. But 
to be really accurate, we'd need to compute the area of a spherical 
cap, not just a plane circle, and there's no point in doing this - 
we're close enough!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/