When the Portland Head Light in Cape Elizabeth, Maine,
was completed in 1791, the tower of this lighthouse stood
72 feet.
By 1865, the tower was raised 20 feet.
How much further out to sea could the beam be seen?
Assume that the Earth is a sphere of radius 4000 miles.
Portland Head
Solution:
In the diagram below, DC represents the lighthouse.
AB and AD represent the radius of the Earth.
There is a right angle at B, the point of tangency.
Solve for BC, the distance from the top of the
lighthouse to the horizon.
Using the Pythagorean Theorem, we find that the distance to the horizon is 10.4446 miles or 55,147 feet when the lighthouse was 72 feet tall, and the distance is 11.806 miles or 62,338 feet when the lighthouse was 92 feet.
Therefore, the beam of light could be seen 1.36 miles further or 7,191 feet.
Walt Arrison sent in the following comments, which I thought would be of interest to many of you:
The accepted formula for determining the horizon
(For practical purposes), used by the Coastal and Geodetic Survey,
US Army Corp of Engineers, US Coast Guard, etc. is miles
squared x .67 feet. Using this formula I came up with 11.77 miles
and 10.39 miles. Also using that formula, the heights would be 93.45 ft
and 73.2 ft. As we surveyors say, " Close enough for government work".
I learned this formula when I first got into the surveying business many,
many years ago. This was long before GPS, electronic equiptment, or any
of the other "stuff" that we use today.
Correctly solved by:
| 1. Walt Arrison | Philadelphia, Pennsylvania |
| 2. Richard K. Johnson | La Jolla, California |
| 3. John Funk | Ventura, California |
| 4. Keith Mealy | Cincinnati, Ohio |
| 5. Rick Jones | Kennett Square, Pennsylvania |
| 6. Rich Murray | Ridgetown, Ontario |
| 7. James Alarie | University of Michigan -- Flint |
| 8. Christopher March | Virginia Tech, Blacksburg |
| 9. ---------- | United Kingdom |
| 10. David Powell | Winchester, Virginia |