Land, Ho!

If you were out in the middle of Lake Erie, at its widest point, you would not be able to see either shoreline because of the curvature of the earth. How tall of a mast would you need on your ship in order to shimmy to the top and be able to see both shorelines? How about for Lake Superior?

Answer:
545 feet high for Lake Erie. 4347 feet high (approx. 4/5 mile) for Lake Superior.

Calculation:

If A and B are the shorelines of a body of water on the surface of the earth, then to overcome the curvature of the earth and see the shoreline, one would need to be at least as high off the surface as the point D, which height is represented by the red arrow. The lines intersecting at D are normal to the surface of the earth at points A and B. As can be seen by observation, the distance represented by the red arrow is simply the difference between R, the radius of the earth, and L, the length of the hypotenuse of a right triangle BCD. The distance R, as well as the circumference of the earth are known and shown in the diagram. If we can determine the angle y, we can solve the triangle and determine the length L.

The width of Lake Erie at its widest point is 92 km, and for Lake Superior is 260 km.

The proportion of the length of the arc AB to the circumference of the earth, 40,008 km, is the proportion of the angle at C of the triangle ABC to the 360 degrees of a circle, so the angle y would be half of this. Therefore, for Lake Erie, which is 92 km wide, the angle y is ((92/40,008) x 360) / 2 = .413917 degrees. Using trigonometry, the cosine of .413917 degrees is the distance R divided by L. The cosine of .413917 is 0.99997391. Therefore, cosine(y) = R/L. L=R/cosine(y) = 6371/.99997391 = 6371.166 km. So the distance represented by the arrow is 6371.166 - 6371 = 0.166 km. There are 3281 feet in a kilometer, so 0.166 x 3281 = 545 feet.

Substituting in the values for Lake Superior we get 4347 feet (about 4/5 of a mile). Just don't ask me to shimmy up there with you!