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### 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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Ratio and Proportion
Middle School Triangles and Other Polygons

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