Can You See Across Lake Michigan?

If you go to the Sears Willis Tower, can you see Michigan? We can use geometry to answer the question.

We can use the Pythagorean Theorem to find the length of the side of a right triangle: a^2+b^2=c^2. In this case the right triangle has sides of d and r and a hypotenuse of r+s. Therefore, the Pythagorean formula is r^2+d^2=(r+s)^2, where r is the radius of the Earth, s is the height of the Sears Tower, and d is the distance one is able to see.

Using FOIL to expand (r+s)^2 gives:

r^2+d^2=r^2+2rs+s^2

 

Since r^2 is on both sides, we can eliminate it.

d^2=2rs+s^2

d=\sqrt{2rs+s^2}.

The radius of the Earth is r=3,963 miles and the height of the Sears Tower in miles is s=0.275379 miles. Consequently, d=46.72 miles. The distance across Lake Michigan is just over 50 miles, so NO, we cannot see all the way to Michigan.

Note: This example assumes that one can only see in a straight line-of-sight. A mirror would allow you to see much farther. In fact, the sky, clouds, and sunlight can form a mirror, a meteorological phenomenon called refraction, and it is sometimes possible to see across Lake Michigan.

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