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/
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