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The "Straightest" Line

When particle linear accelerators (or colliders) are built, a big geometrical problem is faced. That is, the linear acceleration track has to be "perfectly" straight line.

For example, the Stanford linear accelerator for electrons has a three-kilometer long track. To construct this length on the earth's surface, compensations must be made for the curvature of the earth. Thus, special supports are incorporated in the design and construction.

Your Task: Assuming that the middle of the Stanford linear accelerator lies on the surface of the earth, determine the longest special supports needed at the two ends, each one mile from the center.

Note: If you prefer, you could bury the linear accelerato in a tunnel with its two ends on the surface of the earth. Then, you task becomes calculation of how deep the middle of the tunnel will be below the surface of the earth.

Source: Adapted from Reader's Digest Book of Facts, 1987, p. 220.